Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

Abstract

The aims of this paper are to answer several conjectures and questions about the multiplier spectrum of rational maps and giving new proofs of several rigidity theorems in complex dynamics by combining tools from complex and non-Archimedean dynamics.

A remarkable theorem due to McMullen asserts that, aside from the flexible Lattès family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston’s rigidity theorem for post-critically finite maps, in which Teichmüller theory is an essential tool. We will give a new proof of McMullen’s theorem (and therefore a new proof of Thurston’s theorem) without using quasiconformal maps or Teichmüller theory.

We show that, aside from the flexible Lattès family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen’s theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik.

We show that a rational map is exceptional if and only if one of the following holds: (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field, and (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor.

1 Introduction

1.1 Exceptional endomorphisms

Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ of degree at least $2$ . It is called Lattès if it is semi-conjugate to an endomorphism on an elliptic curve. Further, it is called flexible Lattès if it is semi-conjugate to the multiplication by an integer n on an elliptic curve for some $|n|\geq 2.$ We say that f is of monomial type if it is semi-conjugate to the map $z\mapsto z^n$ on ${\mathbb P}^1$ for some $|n|\geq 2.$ We call f exceptional if it is Lattès or of monomial type. An endomorphism f is exceptional if and only if some iterate $f^n$ is exceptional. Exceptional endomorphisms are considered as the exceptional examples in complex dynamics.

In this paper, we will prove a criterion for an endomorphism being exceptional via the information of a homoclinic orbit of f. See Theorem 2.11 for the precise statement, and see Section 2 for the definition and basic properties of homoclinic orbits. Since every f has plenty of homoclinic orbits, the above criterion turns out to be very useful. A direct consequence is the following characterization of exceptional endomorphisms by the linearity of a conformal expending repeller Strategy of the proof of Theorem (CER).

Theorem 1.1. Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ . Assume that f has a linear CER that is not a finite set. Then, f is exceptional.

CER is an impotent concept in complex dynamics introduced by Sullivan [Sul86]. See Section 7.1 for its definition and basic properties.

1.2 Rigidity of stable algebraic families

For $d\geq 1,$ let $\text {Rat}_d({\mathbb C})$ be the space of degree d endomorphisms on ${\mathbb P}^1({\mathbb C})$ . It is a smooth quasi-projective variety of dimension $2d+1$ [Sil12]. Let $FL_d({\mathbb C})\subseteq \text {Rat}_d({\mathbb C})$ be the locus of flexible Lattès maps, which is Zariski closed in $\text {Rat}_d({\mathbb C})$ . The group $\mathrm {PGL}\,_2({\mathbb C})= \mathrm {Aut}({\mathbb P}^1({\mathbb C}))$ acts on $\text {Rat}_d({\mathbb C})$ by conjugacy. The geometric quotient

$$ \begin{align*}{\mathcal M}_d({\mathbb C}):=\text{Rat}_d({\mathbb C})/\mathrm{PGL}\,_2({\mathbb C})\end{align*} $$

is the (coarse) moduli space of endomorphisms of degree d [Sil12]. The moduli space ${\mathcal M}_d({\mathbb C})=\mathrm {Spec \,} ({\mathcal O}(\text {Rat}_d({\mathbb C})))^{\mathrm {PGL}\,_2({\mathbb C})}$ is an affine variety of dimension $2d-2$ [Sil07, Theorem 4.36(c)]. Let $\Psi : \text {Rat}_d({\mathbb C})\to {\mathcal M}_d({\mathbb C})$ be the quotient morphism.

An irreducible algebraic family $f_{\Lambda }$ (of degree d endomorphisms) is an algebraic endomorphism $f_{\Lambda }: {\mathbb P}^1_{\Lambda }\to {\mathbb P}^1_{\Lambda }$ over an irreducible variety $\Lambda $ , such that for every $t\in \Lambda ({\mathbb C})$ , the restriction $f_t$ of $f_{\Lambda }$ above t has degree $d.$ Giving an algebraic family over $\Lambda $ is the same as giving an algebraic morphism $\Lambda \to \text {Rat}_d.$ A family $f_{\Lambda }$ is called isotrivial if $\Psi (\Lambda )$ is a single point.

For every $f\in \text {Rat}_d({\mathbb C})$ and $n\geq 1$ , $f^n$ has exactly $N_n:=d^n+1$ fixed points counted with multiplicity. Their multipliers define a point $s_n(f)\in {\mathbb C}^{N_n}/S_{N_n}$ ,¹ where $S_{N_n}$ is the symmetric group which acts on ${\mathbb C}^{N_n}$ by permuting the coordinates. The multiplier spectrum of f is the sequence $s_n(f), n\geq 1.$ An irreducible algebraic family is called stable if the multiplier spectrum of $f_t$ does not depend on $t\in \Lambda ({\mathbb C})$ .²

In 1987, McMullen [McM87] established the following remarkable characterization of stable irreducible algebraic families.

Theorem 1.2 (McMullen)

Let $f_{\Lambda }$ be a non-isotrivial stable irreducible algebraic family of degree $d\geq 2$ . Then, $f_t\in FL({\mathbb C})$ for every $t\in \Lambda ({\mathbb C})$ .

McMullen’s proof relies on the following Thurston’s rigidity theorem for post-critically finite (PCF) maps [DH93], in which Teichmüller theory is essentially used. An endomorphism f of degree $d\geq 2$ is called PCF if the critical orbits of f are a finite set.

Theorem 1.3 (Thurston)

Let $f_{\Lambda }$ be a non-isotrivial irreducible algebraic family of PCF maps. Then, $f_t\in FL({\mathbb C})$ for every $t\in \Lambda ({\mathbb C})$ .

In this paper, we will give a new proof of McMullen’s theorem without using quasiconformal maps or Teichmüller theory. Since an irreducible algebraic family of PCF maps is automatically stable, this leads to a new proof of Theorem 1.3 without using quasiconformal maps or Teichmüller theory. Except Theorem 2.11, whose proof relies on some basic complex analysis, our proof of Theorem 1.2 only requires some basic knowledge in Berkovich dynamics and hyperbolic dynamics. We explain our strategy of the proof as follows.

Cutting by hypersurfaces, one may reduce to the case that $\Lambda $ is a smooth affine curve. Let W be the smooth projective compactification of $\Lambda $ , and let $B:=W\setminus \Lambda $ . For every $o\in B$ , our family induces a non-Archimedean dynamical system on the Berkovich projective line (see Section 4 for details), which encodes the asymptotic behavior of $f_t$ when $t\to o.$ Since $f_{\Lambda }$ is non-isotrivial and stable, via the study of non-Archimedian dynamics, we show that there is one point $o\in B$ such that when $t\to o$ , $f_t$ ‘degenerates’ to a map taking form $z\mapsto z^m$ in a suitable coordinate, where $2\leq m\leq d-1.$ The above degeneration $z\mapsto z^m$ is called a rescaling limit of $f_{\Lambda }$ at o, in the sense of Kiwi [Kiw15] (see Definition 5.4). On the central fiber, it is easy to find a homoclinic orbit satisfying the condition in our exceptional criterion Theorem 2.11. Using an argument in hyperbolic dynamics [Jon98] (see Lemma 6.1), we can deform such homoclinic orbit to nearby fibers preserving the required condition. By Theorem 2.11, $f_t$ is exceptional for all t sufficiently close to o. We conclude the proof by the fact that exceptional endomorphisms that are not flexible Lattès are isolated in the moduli space ${\mathcal M}_d({\mathbb C})$ .

1.3 Length spectrum as moduli

According to the Noetheriality of the Zariski topology on $\text {Rat}_d$ , McMullen’s rigidity theorem can be reformulated as follows.

Theorem 1.4 (Multiplier spectrum as moduli=Theorem 1.2)

Aside from the flexible Lattès family, the multiplier spectrum determines the conjugacy class of endomorphisms in $\text {Rat}_d({\mathbb C})$ , $d\geq 2$ , up to finitely many choices.

Replace the multipliers by its norm in the definition of multiplier spectrum, and one get the definition of the length spectrum. More precisely, for every $f\in \text {Rat}_d({\mathbb C})$ and $n\geq 1$ , we denote by $L_n(f)\in {\mathbb R}^{N_n}/S_{N_n}$ the element corresponding to the multiset $\{|\lambda _1|,\dots , |\lambda _{N_n}|\},$ where $\lambda _i, i=1,\dots , N_n$ are the multipliers of all $f^n$ -fixed points. The length spectrum of f is defined to be the sequence $L_n(f), n\geq 1.$ A priori, the length spectrum contains less information than the multiplier spectrum. But in this paper, we will show that it determines the conjugacy class up to finitely many choices, therefore generalizing Theorem 1.4.

Theorem 1.5 (Length spectrum as moduli)

Aside from the flexible Lattès family, the length spectrum determines the conjugacy class of endomorphisms in $\text {Rat}_d({\mathbb C})$ , $d\geq 2$ , up to finitely many choices.

1.3.1 Strategy of the proof of Theorem 1.5

Let $g\in \text {Rat}_d({\mathbb C})\setminus FL_d({\mathbb C})$ . We need to show that the image of

$$ \begin{align*} Z:=\{f\in \text{Rat}_d({\mathbb C})\setminus FL_d({\mathbb C}) |\,\, L(f)=L(g)\} \end{align*} $$

in ${\mathcal M}_d({\mathbb C})$ is finite. For $n\geq 0$ , set

$$ \begin{align*} Z_n:=\{f\in \text{Rat}_d({\mathbb C})\setminus FL_d({\mathbb C}) |\,\, L_i(f)=L_i(g), i=1,\dots,n \}. \end{align*} $$

It is clear that $Z_i, i\geq 1$ is a decreasing sequence of closed subsets of $\text {Rat}_d({\mathbb C})\setminus FL_d({\mathbb C})$ and $Z=\cap _{n\geq 1} Z_n.$ For simplicity, we assume that all periodic points of g are repelling. Otherwise, instead of the length spectrum $L(g)$ of all periodic points, we consider the length spectrum $RL(g)$ of all repelling periodic points. Such a change only adds some technical difficulties. To get a contradiction, we assume that $\Psi (Z)\in {\mathcal M}_d({\mathbb C})$ is infinite. Our proof contains two steps.

In Step 1, we show that $Z=Z_N$ for some $N\geq 0.$ We first look at the analogue of this step for the multiplier spectrum. The analogue of $Z_n$ is

$$ \begin{align*} Z^{\prime}_n:=\{f\in \text{Rat}_d({\mathbb C})\setminus FL_d({\mathbb C}) |\,\, s_i(f)=s_i(g), i=1,\dots,n \}, \end{align*} $$

which is Zariski closed in $\text {Rat}_d({\mathbb C})\setminus FL_d({\mathbb C}).$ Hence, $Z_n'$ is stable when n is large by the Noetheriality. This is how Theorem 1.2 implies Theorem 1.4. In the length spectrum case, since the n-th length map $L_n: \text {Rat}_d({\mathbb C})\to {\mathbb R}^{N_n}/S_{N_n}$ takes only real values, it is more natural to view $\text {Rat}_d({\mathbb C})$ as a real algebraic variety by splitting the complex variable into two real variables via $z=x+iy$ . If all $Z_n, n\geq 1$ are real algebraic, we can still conclude this step by the Noetheriality. Unfortunately, this is not true in general (c.f. Theorem 8.10). Since the map $L_n^2$ sending f to $\{|\lambda _1|^2,\dots , |\lambda _{N_n}|^2\}\in {\mathbb R}^{N_n}/S_{N_n}$ is semialgebraic, all $Z_n, n\geq 1$ are semialgebraic. The problem is that, in general, closed semialgebraic sets do not satisfy the descending chain condition. We solve this problem by introducing a class of closed semialgebraic sets called admissible subsets. Roughly speaking, admissible subsets are the closed subsets that are images of algebraic subsets under étale morphisms. See Section 8.2 for its precise definition and basic properties. We show that admissible subsets satisfy the descending chain condition. Under the assumption that all periodic points of g are repelling, we can show that all $Z_n$ are admissible. The admissibility is only used to prove Theorem 1.5.

Step 1 implies that $Z=Z_N$ is semialgebraic. Since $\Psi (Z)$ is infinite, there is an analytic curve $\gamma \simeq [0,1]$ contained in Z such that $\Psi (\gamma )$ is not a point. Every $t\in \gamma \subseteq \text {Rat}_d$ defines an endomorphism $f_t$ . After shrinking $\gamma $ , we may assume that $f_0$ is not exceptional.

In Step 2, we show that the multiplier spectrum of $f_t$ does not depend on $t\in \gamma .$ Once Step 2 is finished, we get a contradiction by Theorem 1.4. Since for every $t\in \gamma $ , $L(f_t)=L(g)$ , all periodic points of $f_t$ are repelling. For every repelling periodic point x of $f_0$ , there is a real analytic map $\phi _x: \gamma \to {\mathbb P}^1({\mathbb C})$ such that for every $t\in \gamma $ , $\phi _x(t)$ and x have the same minimal period and the norms of their multipliers are same. Using homoclinic orbits, we may construct a CER $E_0$ of $f_0$ containing $x.$ It is nonlinear by Theorem 1.1. By Lemma 6.1, for t sufficiently small, $E_0$ can be deformed to a CER $E_t$ of $f_t$ containing $\phi _x(t).$ Using Sullivan’s rigidity theorem [Sul86] (Theorem 7.6), we show that $E_0$ and $E_t$ are conformally conjugate. In particular, the multipliers of $\phi _x(t)$ are a constant for t sufficiently small. Since $\gamma $ is real analytic, the multipliers of $\phi _x(t)$ are a constant on $\gamma .$ Since x is arbitrary, all $f_t,t\in \gamma $ have the same multiplier spectrum. This finishes Step 2.

1.3.2 Further discussion

It is interesting to know whether the uniform version of Theorem 1.5 holds.

Question 1.6. Is there $N\geq 1$ depending only on $d\geq 2$ , such that for every $f\in \text {Rat}_d({\mathbb C})\setminus FL_d({\mathbb C})$ ,

$$ \begin{align*}\#\Psi(\{g\in \text{Rat}_d({\mathbb C})\setminus FL_d({\mathbb C})|\,\, L_i(g)=L_i(f), i=1,\dots,N\})\leq N\,?\end{align*} $$

For every $n\geq 0,$ we set

$$ \begin{align*} R_n:=\{(f,g)\in (\text{Rat}_d({\mathbb C})\setminus FL_d({\mathbb C}))^2|\,\, L_i(f)=L_i(g), i=1,\dots, n\} \end{align*} $$

and

$$ \begin{align*}R_n':=\{(f,g)\in (\text{Rat}_d({\mathbb C})\setminus FL_d({\mathbb C}))^2|\,\, s_i(f)=s_i(g), i=1,\dots, n\}.\end{align*} $$

Both of them are decreasing closed subsets of $(\text {Rat}_d({\mathbb C})\setminus FL_d({\mathbb C}))^2.$ Since all $R_n'$ are algebraic subsets of $(\text {Rat}_d({\mathbb C})\setminus FL_d({\mathbb C}))^2$ , the sequence $R_n'$ is stable for n large. This implies that Theorem 1.4 for the multiplier spectrum is equivalent to its uniform version.

If one can show that the sequence $R_n, n\geq 0$ is stable (for example, if one can show that $R_n$ are admissible), then Question 1.6 has a positive answer. But at the moment, we only know that $R_n$ are semialgebraic.

1.4 Marked multiplier and length spectrum

By Theorem 1.5 and 1.4, aside from the flexible Lattès family, the length spectrum (and therefore the multiplier spectrum) determines the conjugacy class of endomorphisms of degree $d\geq 2$ up to finitely many choices. However, by [Sil07, Theorem 6.62], the multiplier spectrum $f\mapsto s(f)$ (and therefore the length spectrum $f\mapsto L(f)$ ) is far from being injective when d large. For this reason, we consider the marked multiplier and length spectrum. We show that both of them are rigid.

Theorem 1.7 (Marked multiplier spectrum rigidity)

Let f and g be two endomorphisms of ${\mathbb P}^1$ over ${\mathbb C}$ of degree at least $2$ such that f is not exceptional. Assume there is a homeomorphism $h:{\mathcal J}(f)\to {\mathcal J}(g)$ such that $h\circ f=g\circ h$ on ${\mathcal J}(f)$ . Then, the following two conditions are equivalent.

1. (i) There is a nonempty open set $\Omega \subset {\mathcal J}(f)$ such that, for every periodic point $x\in \Omega $ , we have $df^n(x)=dg^n(h(x))$ , where n is the period of x;

2. (ii) h extends to an automorphism $h:{\mathbb P}^1({\mathbb C})\to {\mathbb P}^1({\mathbb C})$ such that $h\circ f=g\circ h$ on ${\mathbb P}^1({\mathbb C})$ .

Let $U,V\subset {\mathbb P}^1({\mathbb C})$ be two open sets. A homeomorphism $h:U\to V$ is called conformal if h is holomorphic or antiholomorphic in every connected component of U. Note that a conformal map h is holomorphic if and only if h preserves the orientation of ${\mathbb P}^1({\mathbb C})$ .

Theorem 1.8 (Marked length spectrum rigidity)

Let f and g be two endomorphisms of ${\mathbb P}^1$ over ${\mathbb C}$ of degree at least $2$ such that f is not exceptional. Assume there is a homeomorphism $h:{\mathcal J}(f)\to {\mathcal J}(g)$ such that $h\circ f=g\circ h$ on ${\mathcal J}(f)$ . Then, the following two conditions are equivalent.

1. (i) There is a nonempty open set $\Omega \subset {\mathcal J}(f)$ such that, for every periodic point $x\in \Omega $ , we have $|df^n(x)|=|dg^n(h(x))|$ , where n is the period of x;

2. (ii) h extends to a conformal map $h:{\mathbb P}^1({\mathbb C})\to {\mathbb P}^1({\mathbb C})$ such that $h\circ f=g\circ h$ on ${\mathbb P}^1({\mathbb C})$ .

Note that if $h:\Omega \to h(\Omega )$ is bi-Lipschitz, then it is not hard to show that for n-periodic point $x\in \Omega $ , we have $|df^n(x)|=|dg^n(h(x))|$ . So the above theorem implies that a locally bi-Lipschitz conjugacy can be improved to a conformal conjugacy on ${\mathbb P}^1({\mathbb C})$ .

Combining Theorem 1.7 and $\lambda $ -Lemma [McM16, Theorem 4.1], we get a second proof of Theorem 1.2. This proof does not use Teichmüller theory, but we need to use quasiconformal maps and the highly nontrivial Sullivan’s rigidity theorem, which is a great achievement in thermodynamic formalism.

Remark 1.9. In Theorem 1.8, in general, h can not be extended to an automorphism on ${\mathbb P}^1({\mathbb C})$ . The complex conjugacy $\sigma : z\mapsto \overline {z}$ induces a mark $h:=\sigma |_{{\mathcal J}(f)}: {\mathcal J}(f)\to \overline {{\mathcal J}(f)}={\mathcal J}(\overline {f})$ , preserving the length spectrum. In general, some periodic point of f may have non-real multipliers. Hence, in this case, h cannot be extended to an automorphism on ${\mathbb P}^1({\mathbb C})$ .

Remark 1.10. Theorem 1.8 was proved by Przytycki and Urbanski in [PU99, Theorem 1.9] under the assumptions that both f and g are tame and $\Omega ={\mathcal J}(f)$ . See [PU99, Definition 1.1] for the precise definition of tameness. In [Ree84, Theorem 2], Rees showed that there are endomorphisms having dense critical orbits and therefore, are not tame.

The study of marked length spectrum rigidity has been investigated in various settings in dynamics and geometry.

In one-dimensional real dynamics, marked multiplier spectrum rigidity was proved for expanding circle maps (see Shub-Sullivan [SS85]) and for some unimodal maps (see Martens-de Melo [MdM99] and Li-Shen [LS06]).

In the contexts of geodesic flows on Riemannian manifolds with negative curvature, a long-standing conjecture stated by Burns-Katok [BK85] (and probably considered even before) asserted the rigidity of marked length spectrum (for closed geodesics). The surface case was proved by Otal [Ota90] and by Croke [Cro90] independently. A local version of the Burns-Katok conjecture in any dimension was proved by Guillarmou-Lefeuvre [GL19].

It was also studied in dynamical billiards. We refer the readers to Huang-Kaloshin-Sorrentino [HKS18], Bálint-De Simoi-Kaloshin-Leguil [BDSKL20], De Simoi-Kaloshin-Leguil [DSKL19] and the references therein.

We prove Theorem 1.8 by combining Theorem 1.1 and Sullivan’s rigidity theorem [Sul86] (see Theorem 7.6). More precisely, let o be a repelling fixed point of f. We construct a family C of CERs of f using homoclinic orbits which covers all backward orbits of o. By Theorem 1.1, all of them are nonlinear. We show that their images under h are CERs of g. Applying Sullivan’s rigidity theorem, we get that conformal conjugacies link the CERs in C to their images. Two CERs in C have ‘large’ intersections. Hence, those conformal conjugacies can be patched together. Using this, we get a conformal extension of h to some disk intersecting the Julia set of f. We can further extend it to a global conformal map on ${\mathbb P}^1({\mathbb C}).$

Theorem 1.7 is a simple consequence of Theorem 1.8 and a result of Eremenko-van Strien [EVS11, Theorem 1] about endomorphisms with real multipliers.

1.5 Zdunik’s rigidity theorem

The following rigidity theorem was proved by Zdunik [Zdu90].

Theorem 1.11 (Zdunik)

Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ of degree at least $2$ . Let $\mu $ be the maximal entropy measure, and let $\alpha $ be the Hausdorff dimension of $\mu $ . Then, $\mu $ is absolutely continous with respect to the $\alpha $ -dimensinal Hausdorff measure $\Lambda _{\alpha }$ on the Julia set if and only if f is exceptional.

Zdunik’s proof is divided into two parts. The first part was proved in her previous work [PUZ89, Theorem 6] with Przytycki and Urbanski. Later, she proved the second part (hence Theorem 1.11) in [Zdu90]. In this paper we will give a simple proof of the second part using Theorem 1.1.

1.6 Milnor’s conjectures on multiplier spectrum

As applications of Theorem 2.11 and Theorem 1.1, we prove two conjectures of Milnor proposed in [Mil06].

Theorem 1.12. Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ of degree at least $2$ . Let K be an imaginary quadratic field. Let $O_K$ be the ring of integers of K. If for every $n\geq 1$ and every n-periodic point x of f, $df^n(x)\in O_K$ . Then, f is exceptional.

The inverse of Theorem 1.12 is also true by Milnor [Mil06, Corollary 3.9 and Lemma 5.6]. In fact, the original conjecture of Milnor concerns the case $K={\mathbb Q}$ . Since imaginary quadratic fields exist (e.g., ${\mathbb Q}(i)$ ) and they contain ${\mathbb Q}$ , Theorem 1.12 implies Milnor’s original conjecture.

Some special cases of Milnor’s conjecture for integer multipliers are known before by Huguin:

1. (i) In [Hug22a], the conjecture was proved for quadratic endomorphisms.

2. (ii) In [Hug21], the conjecture was proved for unicritical polynomials. In fact, Huguin proved a stronger statement, which only assumes the multipliers are in ${\mathbb Q}$ (instead of ${\mathbb Z}$ ).

Remark 1.13. In the recent preprint [Hug22b], Huguin reproved and strengthened our Theorem 1.12. In his result, the multipliers are only assumed to be contained in an arbitrary number field. Huguin’s result relies on an arithmetic equidistribution result for small points proved by Autissier [Aut01] and on a characterization of exceptional maps proved by Zdunik [Zdu14].

The following result confirms another conjecture of Milnor in [Mil06].

Theorem 1.14. Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ of degree at least $2$ . Assume there exists $a>0$ such that for every but finitely many periodic point x, $f^n(x)=x$ , we have $|df^n(x)|=a^n$ . Then, f is exceptional.

Remark 1.15. Theorem 1.14 can also be deduced by a minor modification of an argument of Zdunik [Zdu14].

Let x be a n-periodic point of f. The Lyapunov exponent of x is a real number defined by $\frac {1}{n}\log |df^n(x)|$ . We let $\Delta (f)$ be the closure of the Lyapunov exponents of periodic points contained in the Julia set. Combining Theorem 1.14 and results due to Gelfert-Przytycki-Rams-Rivera Letelier [GPR10], [GPRRL13], we get the following description of $\Delta (f)$ when f is nonexceptional. A closed interval in ${\mathbb R}$ is called nontrivial if it is not a singleton.

Corollary 1.16. Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be a nonexceptional endomorphism over ${\mathbb C}$ of degree at least $2$ . Then, $\Delta (f)$ is a disjoint union of a nontrivial closed interval I and a finite set E (possibly empty). Moreover, there are at most $4$ periodic points whose Lyapunov exponents are contained in E, in particular $|E|\leq 4$ .

1.7 Organization of the paper

In Section 2, we prove some basic properties of homoclinic orbits and we prove the fundamental exceptional criterion Theorem 2.11 by using only the information of a homoclinic orbit. In Section 3, we prove Theorem 1.12. In Section 4, we recall some results about dynamics on the Berkovich projective line. In Section 5, we study the rescaling limit via the dynamics on the Berkovich projective line. In Section 6, we give a new proof of McMullen’s theorem (Theorem 1.2) by studying rescaling limits. In Section 7, we recall some results about CER, and we prove Theorem 1.1, Theorem 1.7, Theorem 1.8, Theorem 1.14 and Corollary 1.16. Moreover, we give a new proof of Theorem 1.11 and we give another proof of Theorem 1.2. In Section 8, we prove Theorem 1.5.

2 Homoclinic orbits and applications

For an endomorphism f of ${\mathbb P}^1$ of degree at least $2$ , we denote by $C(f)$ the set of critical points of f and $PC(f):=\cup _{n\geq 1}f^n(C(f))$ the postcritical set. In this section, ${\mathbb P}^1({\mathbb C})$ is endowed with the complex topology.

Let $f: {\mathbb P}^1\to {\mathbb P}^1$ an endomorphism over ${\mathbb C}$ of degree at least $2.$ Let o be a repelling fixed point of f. A homoclinic orbit ³ of f at o is a sequence of points $o_i, i\geq 0$ satisfying the following properties:

1. (i) $o_0=o$ , $o_1\neq o$ and $f(o_{i})=o_{i-1}$ for $i\geq 1;$

2. (ii) $\lim \limits _{i\to \infty } o_i=o$ .

Be aware that $o_i, i\geq 0$ is actually a backward orbit.

The main result of this section is Theorem 2.11, which is a criterion for an endomorphism f being exceptional via the information of a homoclinic orbit. We will state and prove this theorem at the end of this section.

2.1 Linearization domain and good return times

Define a linearization domain of o to be an open neighborhood U of o such that there is an isomorphism $\phi : U\to {\mathbb D}$ sending o to $0$ , which conjugates $f|_{U_0}:U_0\to U$ to the morphism $z\mapsto \lambda z$ via $\phi $ , where $U_0=f^{-1}(U)\cap U$ and $\lambda =df(o)$ . We call such $\phi $ a linearization on U.

Define g to be the morphism $U\to U$ sending z to $\phi ^{-1}(\lambda ^{-1}\phi (z))$ . It is the unique endomorphism of U satisfiying $f\circ g=\mathrm {id}.$

Remark 2.1. By Koenigs’ theorem [Mil11, Theorem 8.2], for every repelling point o, there is always a linearization domain U. For every $r\in (0,1]$ , $\phi ^{-1}({\mathbb D}(0,r))$ is also a linearization domain of o. In particular, the linearization domains of o form a neighborhood system of $o.$

Remark 2.2. Since g is injective, for every $x\in U$ , $f^{-1}(x)\cap U=g(x).$ In particular, if $o_i\in U$ for $i \geq l$ , then $o_i=g^{i-l}(o_l)$ for all $i\geq l.$

The following lemma shows that for every repelling fixed point o, there are many homoclinic orbits.

Lemma 2.3. For every integer $m\geq 0$ and for every $a\in f^{-m}(o)$ , there is a homoclinic orbit $o_i, i\geq 0$ of o such that $o_m=a.$

Proof. Let U be a linearization domain of $o.$ Since preimages of a are dense in the Julia set, there is $l\geq m$ such that $f^{m-l}(a)\cap U\neq \emptyset .$ Pick $o_l\in f^{m-l}(a)\cap U$ and for $i=0,\dots , l$ , set $o_i:=f^{l-i}(o_l).$ Then $o_0=o$ and $o_m=a.$ For $i\geq l+1$ , set $o_i:=g^{i-l}(o_l)$ . Then $o_i, i\geq 0$ is a homoclinic orbit of $o.$

Definition 2.4. Let U be a connected open neighborhood of o such that U is contained in a linearization domain. For $i\geq 0$ , let $U_i$ be the connected component of $f^{-i}(U)$ containing $o_i$ . An integer $m\geq 1$ is called a good return time for the homoclinic orbit and U if

1. (i) $o_i\in U$ for every $i\geq m$ ;

2. (ii) $U_m\subset \subset U$ , and $f^m:U_m\to U$ is an isomorphism between $U_m$ and $U.$

Remark 2.5. If U itself is a linearization domain and m is a good return time, then i is a good return time for all $i\geq m$ . Indeed, one has $U_i=g^{i-m}(U_m)\subset \subset U$ , and $f^i:U_i\to U$ can be writen as $f^m\circ g^{m-i}$ , which is an isomorphism.

Proposition 2.6. The following statements are equivalent:

1. (i) $o_i\not \in C(f)$ for every $i\geq 1$ ;

2. (ii) there is a linearization domain U and an integer $m\geq 1$ which is a good return time of U;

3. (iii) there is a linearization domain U such that, for every connected open neighborhood V of o, $V\subset U$ , there is an integer $m\geq 1$ which is a good return time of V.

In particular, when $o\not \in PC(f)$ , (i) (and therefore (ii) and (iii)) are satisfied.

Proof. We first show (i) is equivalent to (ii). To see that (ii) implies (i), let m be a good return time of U. Then, by the definition of good return time, $o_i\not \in C(f)$ for $i=1,\dots ,m$ . By Remark 2.5, we conclude that $o_i\notin C(f)$ for every $i\geq 1$ . To see that (i) implies (ii), first choose a linearization domain $U_0$ . Let $g:U_0\to U_0$ be the morphism such that $f\circ g=\mathrm {id}$ . Since $\lim \limits _{i\to \infty } o_i=o$ , there is $l\geq 1$ such that $o_i\in U_0$ for $i\geq l.$ Since $o_i\not \in C(f)$ for every $i\geq 1$ , we have $d(f^l)(o_l)\neq 0.$ So there is an open neighborhood W of $o_l$ in $U_0$ such that $f^l(W)\subseteq U_0$ and $f^l|_W$ is injective. Pick a linearization domain of U of o contained in $f^l(W)$ . Set $U_l:=f^{-l}(U) \cap W.$ Since g is attracting, there is $m\geq l$ such that $g^{m-l}(U_l)\subset \subset U.$ We note that $U_m:=f^{-m}(U)\cap U=g^{m-l}(U_l)$ . Hence, $U_m\subset \subset U$ , and $f^m:U_m\to U$ is an isomorphism. This implies (ii).

It is clear that (iii) implies (ii). It remains to show that (ii) implies (iii). Let $l\geq 1$ be a good return time of U. Let $U_i$ (resp. $V_i$ ) be the connected component of $f^{-i}(U)$ (resp. $f^{-i}(V)$ ) for $i\geq 0$ . We have $U_l\subset \subset U$ . Since g is attracting, there is $m\geq l$ such that $g^{m-l}(U_l)\subset \subset V.$ This implies that m is a good return time of V.

2.2 Adjoint sequence of periodic points

Let U be a linearization domain, and let m be a good return time of U. We construct a sequence of periodic points $q_i, i\geq m$ as follows. By Remark 2.5, for every $i\geq m$ , $f^i|_{U_i}: U_i\to U$ is an isomorphism. Since $U_i\subset \subset U$ , the morphism $(f^i|_{U_i})^{-1}: U\to U_i$ is strictly attracting with respect to the hyperbolic metric on U. Hence, it has a unique fixed point $q_i\in U_i$ . Such $q_i$ is the unique i-periodic point of f which is contained in $U_i.$ Indeed, i is the smallest period of $q_i$ , and $q_i$ is repelling. We call such a sequence an adjoint sequence for the homoclinic orbit $o_i, i\geq 0$ with respect to the linearization domain U and the good return time m (we write $(U,m)$ for short). One can say that a sequence of points $q_i, i\geq 0$ is an adjoint sequence of the homoclinic orbit $o_i, i\geq 0$ if $q_i, i\geq m$ is an adjoint sequence for $o_i, i\geq 0$ with respect to some $(U,m).$ It is clear that for every adjoint sequence $q_i, i\geq 0$ of $o_i, i\geq 0$ , $\lim \limits _{i\to \infty } q_i=o.$ The following lemma shows that the adjoint sequences are unique modulo finite terms.

Lemma 2.7. Let $q_i, i\geq 0$ and $q_i', i\geq 0$ be two adjoint sequence for $o_i, i\geq 0$ . Then, there is $l\geq 0$ such that $q_i=q_i'$ for all $i\geq l$ .

Proof. We only need to prove the case where $q_i, i\geq l$ is an adjoint sequence with respect to $(U, l)$ and $q_i', i\geq l'$ is an adjoint sequence with respect to $(U', l').$ Since there is a linearization domain $U"$ such that $U"\subseteq U\cap U'$ , we may assume that $U'\subseteq U.$ After replacing $l,l'$ by $\max \left \{l,l'\right \}$ , we may assume that $l=l'.$ Then, for every $i\geq l$ , $U^{\prime }_i\subseteq U_i.$ Then, both $q_i$ and $q_i'$ are the unique i-periodic point of f in $U_i$ . So $q_i=q_i'$ for $i\geq l.$

2.3 Poincaré’s linearization map

Set $\lambda :=df(o)\in {\mathbb C}$ . Since o is repelling, $|\lambda |>1.$ Let $(U,m)$ be the pair of linearization domain and good return time for $o_i, i\geq 0$ , and let $q_i, i\geq 0$ be an adjoint sequence.

A theorem of Poincaré [Mil11, Corollary 8.12] says that there is a morphism $\psi : {\mathbb C}\to {\mathbb P}^1({\mathbb C})$ such that $\psi |_{{\mathbb D}}$ gives an isomorphism between ${\mathbb D}$ and U and

(2.1) $$ \begin{align} f(\psi(z))=\psi(\lambda z) \end{align} $$

for every $z\in {\mathbb C}.$ In particular, $\psi |_{{\mathbb D}}^{-1}: U\to {\mathbb D}$ is a linearization of f on $U.$ Such a $\psi $ is called a Poincaré map.

The following criterion for exceptional endomorphisms using the Poincaré map $\psi $ is due to Ritt.

Lemma 2.8 [Rit22]

If the Poincaré map $\psi $ is periodic (i.e., there is a $a\in {\mathbb C}^*$ such that $\psi (z+a)=\psi (z)$ for every $z\in {\mathbb C}$ ), then f is exceptional.

Ritt’s theorem can be easily generalized as following.

Lemma 2.9. If there is an affine automorphism $h: {\mathbb C}\to {\mathbb C}$ such that $h(0)\neq 0$ and $\psi \circ h= \psi $ , then f is exceptional.

Proof. Let G be the group of affine automorphisms g of ${\mathbb C}$ satisfying $\psi \circ g=\psi .$ We have $h\in G$ . It takes form $h: z\mapsto az+b$ , $a\in {\mathbb C}^*$ and $b=h(0)\in {\mathbb C}^*.$ For every $z\in {\mathbb C}$ , we have

$$ \begin{align*}\psi(\lambda h(\lambda^{-1}z))=f\psi(h(\lambda^{-1}z))=f\psi(\lambda^{-1}z)=\psi(z).\end{align*} $$

Hence, the automorphism $g: z\mapsto \lambda h(\lambda ^{-1}z)=az+\lambda b$ is contained in $G.$ Then, $T:=h^{-1}\circ g: z\mapsto z+a^{-1}(\lambda -1)b$ is contained in $G.$ Since $b\neq 0$ and $|\lambda |>1$ , T is a nontrivial translation. We conclude the proof by Lemma 2.8.

Set $P:=\lambda ^m \psi |_{{\mathbb D}}^{-1}(o_m)$ and $V:=\lambda ^m(\psi |_{{\mathbb D}}^{-1}(U_m)).$ For $i\geq m,$ set $Q_i:=\psi |_{{\mathbb D}}^{-1}(q_i).$ One has $\psi (V)=U$ , $\psi (P)=o$ , and $\psi |_V: V\to U$ is an isomorphism. We set $T:=(\psi |_V)^{-1}\circ \psi |_{{\mathbb D}}:{\mathbb D}\to V$ . Then T is an isomorphism. Similar constructions of T appeared already in the works of Ritt [Rit22] and Eremenko-van Strien [EVS11]. We summarize our construction in the following figure.

We have $\psi \circ T=\psi $ and $T(0)=P.$ Moreover, by our construction, we have for every $i\geq m$ , $V= \lambda ^i (\psi |_{{\mathbb D}})^{-1}(U_i)$ . In particular, $\lambda ^i Q_i\in V$ . By (2.1) we have

$$ \begin{align*} \psi(\lambda^i Q_i)=f^i(\psi(Q_i))=f^i(q_i)=q_i. \end{align*} $$

This implies

(2.2) $$ \begin{align} T(Q_i)=\lambda^iQ_i. \end{align} $$

Since $\lim \limits _{i\to \infty }q_i=o$ , we have $\lim \limits _{i\to \infty }Q_i=0$ and

(2.3) $$ \begin{align} \lim_{i\to \infty}\lambda^iQ_i=P. \end{align} $$

By (2.1), we have for every $i\geq 1$ ,

(2.4) $$ \begin{align} df^i(\psi(z))d\psi(z)=\lambda^i d\psi(\lambda^i z), \end{align} $$

and by $\psi \circ T=\psi $ , we have

(2.5) $$ \begin{align} d\psi(T(z))T'(z)=d\psi(z). \end{align} $$

Set $z=Q_i$ . Combine (2.2), (2.4) and (2.5), and we get

$$ \begin{align*} df^i(q_i)d\psi(\lambda^i Q_i)T'(Q_i)=\lambda^i d\psi(\lambda^i Q_i). \end{align*} $$

Since zeros of a holomorphic function are isolated, as $\lambda ^i Q_i \to P$ , for i large enough, we have $d\psi (\lambda ^i Q_i)\neq 0$ . Hence, for i large enough,

(2.6) $$ \begin{align} \lambda^iT'(Q_i)^{-1}=df^i(q_i). \end{align} $$

The following observation will be used in the proof of Theorem 1.12.

Lemma 2.10. Set $\theta :=1/T': {\mathbb D}\to {\mathbb C}$ . We have

$$ \begin{align*}\lim_{i\to \infty}(df^i(q_i)-\lambda^i\theta(0))=P\theta'(0).\end{align*} $$

Proof. By (2.3) and (2.6),we have

$$ \begin{align*}\lim_{i\to \infty}(df^i(q_i)-\lambda^i\theta(0))/P=\lim_{i\to \infty}(df^i(q_i)-\lambda^i\theta(0))/\lambda^iQ_i\end{align*} $$

$$ \begin{align*}=\lim_{i\to \infty}(df^i(q_i)/\lambda^i-\theta(0))/Q_i=\lim_{i\to \infty}(\theta(Q_i)-\theta(0))/Q_i=\theta'(0),\end{align*} $$

which concludes the proof.

The following is the main result of this section, which characterizes exceptional endomorphisms by using the multipliers of adjoint sequence of a homoclinic orbit.

Theorem 2.11. Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ of degree at least $2$ . Let o be a repelling fixed point of f such that $df(o)=\lambda $ . Let $o_i, i\geq 0$ be a homoclinic orbit of o such that $o_i\notin C(f)$ for every $i\geq 0$ . Assume that there is $C\in {\mathbb C}^\ast $ , such that for one (and therefore, every) adjoint sequence $q_i, i\geq 0$ of $o_i, i\geq 0$ , $df^i(q_i)=C\lambda ^i$ for i large. Then f is exceptional.

Proof. We may assume that $q_i, i\geq m$ is adjoint with respect to the linearization domain and good return time $(U,m)$ for $o_i, i\geq 0$ , and $d(f^i)(q_i)=C\lambda ^i$ for all $i\geq m.$ By (2.6), we get $T'(Q_i)=C^{-1}$ for $i\geq m.$ Since $Q_i\neq 0$ for $i\geq m$ and $\lim \limits _{i\to \infty }Q_i=0$ , $T'=C^{-1}$ on ${\mathbb D}$ . It follows that $T(z)=C^{-1}z+P$ for every $z\in {\mathbb D}.$ Then, T extends to the affine endomorphism on ${\mathbb C}$ sending z to $C^{-1}z+P.$ One has $\psi =\psi \circ T$ on ${\mathbb C}.$ We conclude the proof by Lemma 2.9.

3 Proof of Milnor’s conjecture

In this section, we prove one of Milnor’s conjectures (Theorem 1.12). We postpone the proof of another conjecture of Milnor (Theorem 1.14) to Section 7.

Proof of Theorem 1.12

Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ of degree at least $2$ . Let K be an imaginary quadratic field. Assume that for every $n\geq 1$ and every n-periodic point x of f, $df^n(x)\in O_K.$

After replacing f by a suitable positive iterate, we may assume that f has a repelling fixed point $o\notin PC(f)$ . Let $o_i, i\geq 0$ be a homoclinic orbit of $o.$ By Proposition 2.6, there is a linearization domain and a good return time $(U,m)$ for $o_i, i\geq 0$ . Let $q_i, i\geq m$ be the adjoint sequence for it. Set $\mu _i:=df^i(q_i)\in O_K$ for $i\geq m$ . Set $\lambda :=df(o).$

Lemma 3.1. There are $a\in K^*,b\in K$ such that $\mu _i=a\lambda ^i+b$ for i large.

Proof of Lemma 3.1

We view K as a subfield of ${\mathbb C}.$ Then, $O_K$ is a discrete subgroup of $({\mathbb C}, +)$ . Set ${\mathbb T}:={\mathbb C}/O_K$ and $\pi : {\mathbb C}\to {\mathbb T}$ the quotient map. Since $\lambda \in O_K$ , the multiplication by $\lambda $ on L descends to an endomorphism $[\lambda ]$ on ${\mathbb T}.$ By Lemma 2.10, we have

(3.1) $$ \begin{align}\lim_{i\to \infty}(\mu_i-a\lambda^i)=b, \end{align} $$

where $a=\theta (0)=1/T'(0)\in {\mathbb C}^*$ and $b=P\theta '(0)\in {\mathbb C}$ (See Section 2 for the definitions of T and $\theta $ ). Since $\mu _i\in O_K, i\geq m$ , we get

$$ \begin{align*}\lim_{i\to \infty}[\lambda]^i\pi(a)=\pi(b).\end{align*} $$

In particular, $\pi (b)$ is fixed by $[\lambda ].$ Since $d[\lambda ](b)=\lambda $ , $[\lambda ]$ is repelling at $\pi (b).$ Hence, for i large, we must have

(3.2) $$ \begin{align} [\lambda]^i\pi(a)=\pi(b). \end{align} $$

Since $O_K$ is discrete in ${\mathbb C}$ , by (3.1) and (3.2), we have

(3.3) $$ \begin{align}\mu_i=a\lambda^i+b \text{ for } i \text{ large}. \end{align} $$

There are $n> l\geq m$ such that $\mu _n=a\lambda ^n+b$ and $\mu _l=a\lambda ^l+b$ . This implies that $a,b\in K.$

After enlarging $m,$ we may assume that $\mu _i=a\lambda ^i+b$ for all $i\geq m.$ Assume by contradiction that f is not exceptional. By Theorem 2.11, we must have $b\neq 0.$ For ${\mathbf {p}}\in \mathrm {Spec \,} O_K$ , let $K_{\mathbf {p}}$ be the completion of K with respect to ${\mathbf {p}}$ . Denote by $|\cdot |_{\mathbf {p}}$ the ${\mathbf {p}}$ -adic norm on $K_{\mathbf {p}}$ normalized by $|p|_{{\mathbf {p}}}=p^{-1}$ where $p:=\mathrm {char\,} O_K/{\mathbf {p}}.$ Let $K_{\mathbf {p}}^{\circ }$ be the valuation ring of $K_{{\mathbf {p}}}$ . For $\mu \in O_K$ , $\mu \in {\mathbf {p}}$ if and only if $|\mu |_{\mathbf {p}}<1$ .

Lemma 3.2. For ${\mathbf {p}}\in \mathrm {Spec \,} O_K$ and $\epsilon>0$ , if $\lambda \not \in {\mathbf {p}}$ , then there is $N\in {\mathbb Z}_{>0}$ such that $|\lambda ^{Ni}-1|_{\mathbf {p}}<\epsilon $ for all $i\geq 0.$

Proof of Lemma 3.2

Since $O_K/{\mathbf {p}}$ is a finite field and $\lambda \not \in {\mathbf {p}}$ , there is $l\geq 1$ such that $\lambda ^l-1\in {\mathbf {p}}.$ Since

$$ \begin{align*}\lim_{n\to \infty}\lambda^{lp^n}=\lim_{n\to \infty}(1+(\lambda^l-1))^{p^n}=1\end{align*} $$

in the ${\mathbf {p}}-$ adic topology, there is $N\in {\mathbb Z}_{>0}$ , such that $|\lambda ^N-1|_{{\mathbf {p}}}<\epsilon .$ Then, for every $i\geq 0$ , $|\lambda ^{Ni}-1|_{{\mathbf {p}}}=|\lambda ^{N}-1|_{{\mathbf {p}}} |1+\lambda ^N\dots +\lambda ^{N(i-1)}|_{{\mathbf {p}}}<\epsilon .$

Let S be the finite set of prime ideals ${\mathbf {p}}\in \mathrm {Spec \,} O_K\setminus \{0\}$ dividing $\lambda (\deg f)!\in O_K.$ For every ${\mathbf {p}}\in \mathrm {Spec \,} O_K\setminus (S\cup \{0\})$ , there is an embedding of field $\tau _K: K\hookrightarrow {\mathbb C}_p$ such that $|\cdot |_{{\mathbf {p}}}$ is the restriction of the norm on ${\mathbb C}_p$ via this embedding. Recall that ${\mathbb C}_p$ is the completion of the algebraic closure of ${\mathbb Q}_p$ . Then, $\tau _K$ extends to an isomorphism $\tau :{\mathbb C}\to {\mathbb C}_p$ . Via $\tau $ , the norm $|\cdot |_{{\mathbf {p}}}$ extends to a non-Archimedean complete norm on ${\mathbb C}$ . By [RL03a, Corollaire 4.7 and Corollaire 4.9] of Rivera-Letelier (or [BIJL14, Corollary 1.6] of Benedetto-Ingram-Jones-Levy), for every ${\mathbf {p}}\in \mathrm {Spec \,} O_K\setminus (S\cup \{0\})$ , there are at most finitely many integers $i\geq m$ satisfying $|\mu _i|_{\mathbf {p}}<1$ . We claim that for every $i\geq m,$ we have $\mu _i=a\lambda ^i+b\not \in {\mathbf {p}}$ for every ${\mathbf {p}}\in \mathrm {Spec \,} O_K\setminus (S\cup \{0\})$ . In fact if there is ${\mathbf {p}}\in \mathrm {Spec \,} O_K\setminus (S\cup \{0\})$ such that $a\lambda ^i+b\in {\mathbf {p}}$ for some $i\geq m$ , by Lemma 3.2, there is $N\in {\mathbb Z}_{>0}$ , such that for all $j\geq 0$ , $|\lambda ^{Nj}-1|_{\mathbf {p}}<|a^{-1}|/2.$ Then, for every $j\geq m$ , we get

$$ \begin{align*}|\mu_{i+Nj}|_{{\mathbf{p}}}=|a\lambda^{i+Nj}+b|_{{\mathbf{p}}}\leq \max\{|a\lambda^{i}+b|_{{\mathbf{p}}}+|a\lambda^i(\lambda^{Nj}-1)|_{{\mathbf{p}}}\}<1.\end{align*} $$

Thus, we obtain infinitely many integers $i\geq m$ satisfying $|\mu _i|_{\mathbf {p}}<1,$ which is a contradiction.

Set $S':=\{{\mathbf {p}}\in S|\,\, \lambda \in {\mathbf {p}}\}$ and $S"=S\setminus S'.$ Since $a\neq 0$ , there is $l\geq 0$ such that $a\lambda ^l+b\neq 0$ . Set

$$ \begin{align*}A:=\min(\{|a\lambda^{l}+b|_{{\mathbf{p}}}|\,\, {\mathbf{p}}\in S"\}\cup \{|b|_{{\mathbf{p}}}|\,\, {\mathbf{p}}\in S'\}\cup \{1\})>0.\end{align*} $$

For every ${\mathbf {p}}\in S'$ , there is an integer $M_{{\mathbf {p}}}\geq m$ such that $|a\lambda ^{M_{{\mathbf {p}}}}|_{{\mathbf {p}}}<|b|_{{\mathbf {p}}}.$ Then, for every $i\geq M_{{\mathbf {p}}}, {\mathbf {p}}\in S'$ , we have

$$ \begin{align*}|\mu_i|_{{\mathbf{p}}}=|b|_{{\mathbf{p}}}\geq A.\end{align*} $$

For every ${\mathbf {p}}\in S"$ , by Lemma 3.2, there is $N_{{\mathbf {p}}}\in {\mathbb Z}_{>0}$ such that for every $j\geq 0$ , $|\lambda ^{N_{{\mathbf {p}}}j}-1|_{{\mathbf {p}}}<|a^{-1}|_{{\mathbf {p}}}|a\lambda ^{l}+b|_{{\mathbf {p}}}.$ Then, for all $j\geq m$ , we have

$$ \begin{align*}|\mu_{l+N_{{\mathbf{p}}}j}|_{{\mathbf{p}}}=|a\lambda^{l+N_{{\mathbf{p}}}j}+b|_{{\mathbf{p}}}=|(a\lambda^{l}+b)+a\lambda^i(\lambda^{N_{{\mathbf{p}}}j}-1)|_{{\mathbf{p}}}=|a\lambda^{l}+b|_{{\mathbf{p}}}\geq A.\end{align*} $$

Set $M:=\max \{M_{{\mathbf {p}}}|\,\, {\mathbf {p}}\in S'\}$ and $N:=\prod _{{\mathbf {p}}\in S"}N_{{\mathbf {p}}}.$ For every $i\geq M$ , by the above discussion, we get $|\mu _{l+Ni}|_{{\mathbf {p}}}\geq A$ for all ${\mathbf {p}}\in S$ . Fix an embedding of K in ${\mathbb C}$ . For every ${\mathbf {p}}\in \mathrm {Spec \,} O_K\setminus \{0\},$ set $n_{{\mathbf {p}}}:=[K_{{\mathbf {p}}}:{\mathbb Q}_{p}]$ with $p=\mathrm {char\,} O_K/{\mathbf {p}}.$ We have $n_{\mathbf {p}}\leq 2$ . By product formula, we get, since $|\mu _{l+Ni}|_{{\mathbf {p}}}=1$ for all ${\mathbf {p}}\in \mathrm {Spec \,} O_K\setminus (S\cup \{0\})$ ,

$$ \begin{align*}|\mu_{l+Ni}|^{[K:{\mathbb Q}]}=\prod_{{\mathbf{p}}\in \mathrm{Spec \,} O_K\setminus \{0\}}|\mu_{l+Ni}|_{{\mathbf{p}}}^{-n_{{\mathbf{p}}}}=\prod_{{\mathbf{p}}\in S}|\mu_{l+Ni}|_{{\mathbf{p}}}^{-n_{{\mathbf{p}}}}\leq A^{-2|S|},\end{align*} $$

where $i\geq m$ .

Hence, $\mu _{l+Ni}, i\geq m$ is bounded in ${\mathbb C}.$ Since $a\neq 0$ and $|\lambda |>1$ , we get a contradiction. The proof is finished.

4 The Berkovich projective line

Let ${\mathbf {k}}$ be a complete valued field with a nontrivial non-Archimedean norm $|\cdot |$ . We denote by ${\mathbf {k}}^{\circ }$ the valuation ring of ${\mathbf {k}}$ , ${\mathbf {k}}^{\circ \circ }$ the maximal ideal of ${\mathbf {k}}^{\circ }$ and $\tilde {k}={\mathbf {k}}^{\circ }/{\mathbf {k}}^{\circ \circ }$ the residue field.

In this section, we collect some basic facts about Berkovich’s analytification of ${\mathbb P}^1_{{\mathbf {k}}}$ . We refer the readers to [Ber90] for a general discussion on Berkovich space, and to [BR10] for a detailed description of the Berkovich projective line and the dynamics on it.

4.1 Analytification of the projective line

Let ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ be the analytification of ${\mathbb P}^{1}_{{\mathbf {k}}}$ in the sense of Berkovich, which is a compact topological space endowed with a structural sheaf of analytic functions. Only its topological structure will be used in this paper. We describe it briefly below.

The analytification ${\mathbb A}^{1,\mathrm {an}}_{{\mathbf {k}}}$ of the affine line ${\mathbb A}^1_{{\mathbf {k}}}$ is the space of all multiplicative semi-norms on ${\mathbf {k}}[z]$ whose restriction to ${\mathbf {k}}$ coincide with $|\cdot |$ , endowed with the topology of pointwise convergence. For any $x\in {\mathbb A}^{1,\mathrm {an}}_{{\mathbf {k}}}$ and $P\in {\mathbf {k}}[z]$ , it is customary to denote $|P(x)|:=|P|_x$ , where $|\cdot |_x$ is the semi-norm associated to x.

As a topological space, ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ is the one-point compactification of ${\mathbb A}_{{\mathbf {k}}}^{1,\mathrm { an}}$ . We write ${\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}={\mathbb A}_{{\mathbf {k}}}^{1,\mathrm {an}}\cup \{\infty \}$ . More formally, it is obtained by gluing two copies of ${\mathbb A}_{{\mathbf {k}}}^{1,\mathrm {an}}$ in the usual way via the transition map $z\mapsto z^{-1}$ on the punctured affine line $({\mathbb A}_{{\mathbf {k}}}^1\setminus \{0\})^{\mathrm {an}}$ .

The Berkovich projective line ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ is an ${\mathbb R}$ -tree in the sense that it is uniquely path-connected (see [Jon15, Section 2] for the precise definitions). In particular, for $x,y\in {\mathbb P}^{1,\mathrm { an}}_{{\mathbf {k}}}$ , there is a well-defined segment $[x,y]$ .

For $a\in {\mathbf {k}}$ and $r\in [0,+\infty )$ , we denote ${\mathbb D}(a,r)$ by the closed disk ${\mathbb D}(a,r):=\{x\in {\mathbb A}^{1,\mathrm { an}}_{{\mathbf {k}}}:\, |(z-a)(x)|\leq r\}.$ One may check that the norm $\sum _{i\geq 0}a_i(z-a)^i\mapsto \max \{|a_i|r^i, i\geq 0\}$ defines a point $\xi _{a,r}\in {\mathbb D}(a,r).$ One may set $x_{G}:=\xi _{0,1}$ and call it the Gauss point.

Remark 4.1. When $r=0$ , $\xi _{a,0}$ is exactly the image of a via the identification ${\mathbf {k}}={\mathbb A}^1({\mathbf {k}})\hookrightarrow {\mathbb A}_{{\mathbf {k}}}^{1,\mathrm {an}}.$

The group $\mathrm {PGL}\,_2({\mathbf {k}})$ acts on ${\mathbb P}^1_{{\mathbf {k}}}$ , and therefore on ${\mathbb P}^{1, \mathrm {an}}_{{\mathbf {k}}}$ .

Lemma 4.2. [DF19, Proposition 1.4] For a point $x\in {\mathbb P}^{1, \mathrm {an}}_{{\mathbf {k}}}$ , $x\in \mathrm {PGL}\,_2({\mathbf {k}})\cdot x_g$ if and only if it takes form $x=\xi _{a,r}$ for some $a\in {\mathbf {k}}$ and $r\in |{\mathbf {k}}^*|.$

Remark 4.3. The stablizer of $\mathrm {PGL}\,_2({\mathbf {k}})$ at $x_g$ is $\mathrm {PGL}\,_2({\mathbf {k}}^{\circ })$ , which is open in $\mathrm { PGL}\,_2({\mathbf {k}})$ . So for any dense subfield L of ${\mathbf {k}}$ , we have $\mathrm {PGL}\,_2(L)\cdot x_g=\mathrm {PGL}\,_2({\mathbf {k}})\cdot x_g.$

4.2 Points in ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$

Let $\widehat {\overline {{\mathbf {k}}}}$ be the completion of the algebraic closure of ${\mathbf {k}}$ . It is still algebraically closed. By [Ber90, Corollary 1.3.6], $\mathrm {Aut}(\widehat {\overline {{\mathbf {k}}}}/{\mathbf {k}})$ acts on ${\mathbb P}^{1,\mathrm {an}}_{\widehat {\overline {{\mathbf {k}}}}}$ and we have ${\mathbb P}^1_{\widehat {\overline {{\mathbf {k}}}}}/\mathrm { Aut}(\widehat {\overline {{\mathbf {k}}}}/{\mathbf {k}})= {\mathbb P}^1_{{\mathbf {k}}}.$ We denote by $\pi :{\mathbb P}^{1,\mathrm { an}}_{\widehat {\overline {{\mathbf {k}}}}}\to {\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ the quotient map. The points of ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ can be classified into 4 types:

1. (i) a type 1 point takes form $\pi (a)$ where $a\in \widehat {\overline {{\mathbf {k}}}}\cup \{\infty \}={\mathbb P}^{1,\mathrm { an}}_{\widehat {\overline {{\mathbf {k}}}}}$ ;

2. (ii) a type 2 point takes form $\pi (\xi _{x,r})$ where $x\in \widehat {\overline {{\mathbf {k}}}}$ and $r\in |\widehat {\overline {{\mathbf {k}}}}^*|$ ;

3. (iii) a type 3 point takes form $\pi (\xi _{x,r})$ where $x\in \widehat {\overline {{\mathbf {k}}}}$ and $r\in {\mathbb R}_{>0}\setminus |\hat {\overline {{\mathbf {k}}}}^*|$ ;

4. (iv) a type 4 point takes form $\pi (x)$ where x is the pointwise limit of $\xi _{x_i,r_i}$ such that the corresponding discs ${\mathbb D}(x_i,r_i)$ form a decreasing sequence with empty intersection.

See [Ber90, Section 1.4.4] for further details when ${\mathbf {k}}$ is algebraically closed. See also [Ked11, Proposition 2.2.7] and [Ste19, Section 2.1]. The set of type 1 (resp. type 2) points is dense in ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ . Points of type 4 exist only when ${\mathbf {k}}$ is not spherically complete. If we view ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ as a metric tree, then the end points have type 1 or 4.

For every $x\in {\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ , we can define an equivalence relation on the set ${\mathbb P}^{1,\mathrm { an}}_{{\mathbf {k}}}\setminus \{x\}$ as follows: $y\sim z$ if the paths $(x, y]$ and $(x,z]$ intersect. The tangent space $T_{x}$ at x is the set of equivalences classes of ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}\setminus \{x\}$ modulo $\sim $ . See [Jon15, Section 2.5] for details. If x is an end point (a point of type 1 or 4), then $|T_{x}|=1$ . If x is of type 3, then $|T_{x}|=2$ . If x is of type 2, then $|T_{x}|\geq 3.$ For a direction $v\in T_x$ , let $U(v)$ be the set of all $y\in {\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ such that the path $(x,y]$ presents $v.$ Then, $U(v)$ is an open subset such that $\partial {U(v)}=x$ .

4.3 Dynamics on ${\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}$

Let $f:{\mathbb P}^1_{{\mathbf {k}}}\to {\mathbb P}^1_{{\mathbf {k}}}$ be an endomorphism of degree $d\geq 2.$ We still denote by f the induced endomorphism on ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ .

4.3.1 The tangent map

For $x,y\in {\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}$ , if $f(x)=y$ , then $x,y$ have the same type. Moreover, f induces a tangent map $T_xf: T_x\to T_y$ sending $v\in T_x$ to the unique direction $w\in T_y$ such that for every $z\in U(v)$ , $(y,f(z)]\cap U(w)\neq \emptyset .$ We note that, in general, $f(U(v))$ may not be equal to $U(w).$ If $f(U(v))=U(w)$ , we say that v is a good direction. Otherwise, it is called a bad direction. If v is a bad direction, then $f(U(v))={\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ [Ben, Theorem 7.34].

We may naturally identify $T_{x_{G}}$ with ${\mathbb P}^1(\tilde {{\mathbf {k}}})$ as follows. Consider the standard model ${\mathbb P}^1_{{\mathbf {k}}^{\circ }}$ of ${\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}$ . There is a reduction map $\mathrm {red}: {\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}\to {\mathbb P}^1_{\tilde {{\mathbf {k}}}}.$ The preimage of the generic point of ${\mathbb P}^1_{\tilde {{\mathbf {k}}}}$ is the Gauss point $x_G$ , and for every $y\in {\mathbb P}^1(\tilde {{\mathbf {k}}})$ , there is a unique $v_y\in T_{x_{G}}$ such that $U(v_y)=\mathrm {red}^{-1}(y).$ The map ${\mathbb P}^1(\tilde {{\mathbf {k}}})\to T_{x_{G}}$ sending y to $v_y$ is bijective. Let h be any endomorphism of ${\mathbb P}^1_{{\mathbf {k}}}$ such that $h(x_G)=x_G$ , and it extends to a rational self-map $h_{{\mathbf {k}}^{\circ }}$ of ${\mathbb P}^1_{{\mathbf {k}}^{\circ }}$ . We denote by $\tilde {h}: {\mathbb P}^1_{\tilde {{\mathbf {k}}}}\to {\mathbb P}^1_{\tilde {{\mathbf {k}}}}$ the restriction of h to the special fiber of ${\mathbb P}^1_{{\mathbf {k}}^{\circ }}$ and call it the reduction of $h.$ Then, $T_{x_G}h: T_{x_G}={\mathbb P}^1(\tilde {{\mathbf {k}}}) \to T_{x_G}$ is induced by $\tilde {h}$ . We define $\deg T_{x_G}h$ to be the degree of $\tilde {h}.$ We note that $\deg \tilde {h}\leq \deg h$ . The equality holds if and only if $h_{{\mathbf {k}}^{\circ }}$ is an endomorphism. In this case, we say that h has explicit good reduction.

More generally, for every $x,y\in \mathrm {PGL}\,_2({\mathbf {k}})\cdot x_{G}$ with $f(x)=y$ , we may define

$$ \begin{align*}\deg T_{x}f:=\deg T_{x_G}(h^{-1}fg)=\deg \widetilde{h^{-1}fg},\end{align*} $$

where $h,g\in \mathrm {PGL}\,_2({\mathbf {k}})$ with $g(x_G)=x$ and $h(x_G)=y.$ Then, $1\leq \deg T_{x_G}f\leq \deg f$ and $\deg T_{x_G}f$ does not depend on the choices of $g,h.$

Remark 4.4. Assume that ${\mathbf {k}}$ is algebraically closed. By Lemma 4.2, the set of type 2 points in ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ is exactly $\mathrm {PGL}\,_2({\mathbf {k}})\cdot x_G.$

4.3.2 Periodic points

Assume that ${\mathbf {k}}$ is algebraically closed. For $n\geq 1$ , a n-periodic point of f is a point $x\in {\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ such that $f^n(x)=x.$ They can be divided into three types: attracting, indifferent and repelling. A type 1 periodic point $x\in {\mathbb P}^{1}({\mathbf {k}})$ of period $n\geq 1$ is called attracting if $|d(f^n)(x)|<1$ ; indifferent if $|d(f^n)(x)|=1$ ; and repelling if $|d(f^n)(x)|>1.$ A n-periodic point $x\in {\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}$ of type 2 is called indifferent if $\deg T_{x}f=1$ ; repelling if $\deg T_{x}f\geq 2.$ Every n-periodic point $x\in {\mathbb P}_{{\mathbf {k}}}^{1,\mathrm {an}}$ of type 3 or 4 are indifferent [RL03b, Lemma 5.3, 5.4].

4.3.3 Fatou and Julia sets

Assume that ${\mathbf {k}}$ is algebraically closed.

The Julia set of f is the set ${\mathcal J}(f)$ of points $z \in {\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ with the following property: for every neighborhood U of z, the union of iterates $\bigcup _{n\geq 0} f^n(U)$ omits only finitely many points of ${\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}$ . Its complement ${\mathcal F}(f):={\mathbb P}^{1,\mathrm {an}}_{{\mathbf {k}}}\setminus {\mathcal J}(f)$ is the Fatou set of f.

We list some basic properties of the Julia and Fatou sets of f.

Proposition 4.5 [Ben, Chapter 8 and Section 12.2]

1. (i) The Fatou set ${\mathcal F}(f)$ is open and the Julia set ${\mathcal J}(f)$ is closed.

2. (ii) All attracting periodic points of f are contained in ${\mathcal F}(f)$ .

3. (iii) All repelling periodic points of f are contained in ${\mathcal J}(f)$ .

4. (iv) We have ${\mathcal J}(f)=f({\mathcal J}(f))=f^{-1}({\mathcal J}(f))$ and ${\mathcal F}(f)=f({\mathcal F}(f))=f^{-1}({\mathcal F}(f))$ .

5. (v) Both ${\mathcal J}(f)$ and ${\mathcal F}(f)$ are nonempty.

6. (vi) For every $z\in {\mathcal J}(f)$ , $\cup _{n\geq 0}f^{-n}(z)$ is dense in ${\mathcal J}(f)$ .

7. (vii) Repelling periodic points are dense in ${\mathcal J}(f)$ .

4.3.4 Good reduction

We say f has good reduction if, after some coordinate change $h \in \mathrm {PGL}\,_2({\mathbf {k}})$ , the map $h^{-1} \circ f \circ h $ has explicit good reduction.

Theorem 4.6 [FRL10, Theorem E]

The endomorphism f has explicit good reduction if and only if ${\mathcal J}(f)=x_G$ . Moreover, if ${\mathbf {k}}$ is algebraically closed, f has good reduction if and only if ${\mathcal J}(f)$ is a single point.

Remark 4.7. Assume that ${\mathbf {k}}$ is algebraically closed. If ${\mathcal J}(f)$ is a single point, then by Theorem 4.6 and (vii) of Proposition 4.5, it is a type 2 repelling point.

5 Rescaling limits of holomorphic families

5.1 Holomorphic families

Recall that $\Psi : \text {Rat}_d({\mathbb C})\to {\mathcal M}_d({\mathbb C})$ is the quotient morphism, where ${\mathcal M}_d({\mathbb C}):=\text {Rat}_d({\mathbb C})/\mathrm {PGL}\,_2({\mathbb C})$ is the moduli space.

Let $\Lambda $ be a complex manifold. We denote by ${\mathcal O}^{\mathrm {an}}(\Lambda )$ the ring of holomorphic functions on $\Lambda .$ Moreover, if $\Lambda $ is a complex algebraic variety, we denote by ${\mathcal O}(\Lambda )$ the ring of algebraic functions on $\Lambda .$

A holomorphic (resp. meromorphic) family on $\Lambda $ is an endomorphism (resp. meromorphic self-map) $f_{\Lambda }$ on ${\mathbb P}^1\times \Lambda $ such that $\pi _{\Lambda }\circ f_{\Lambda }=\pi _{\Lambda }$ , where $\pi _{\Lambda }:{\mathbb P}^1({\mathbb C})\times \Lambda \to \Lambda $ is the projection to $\Lambda .$ More concretely, one may write $f_{\Lambda }([x:y],t)=([P_t(x,y):Q_t(x,y)], t)$ where $P_t(x,y), Q_t(x,y)$ are homogenous polynomials of same degree d in ${\mathcal O}^{\mathrm {an}}(\Lambda )[x,y]$ without a common divisor. We say that $f_{\Lambda }$ is of degree d. Then, $f_{\Lambda }$ is holomorphic if there is no $(t,x,y)\in \Lambda \times {\mathbb C}^*\times {\mathbb C}^*$ such that $P_t(x,y)=Q_t(x,y)=0.$

For $t\in \Lambda $ , we denote by $f_t$ the restriction of $f_{\Lambda }$ to the fiber above $t.$ We denote by $I(f_{\Lambda })$ the indeterminacy locus of $f_{\Lambda }$ and $B(f_{\Lambda }):=\pi _{\Lambda }(I(f_{\Lambda })).$ Then, $I(f_{\Lambda })$ and $B(f_{\Lambda })$ are proper closed analytic subspaces of ${\mathbb P}^1\times \Lambda $ and $\Lambda $ , respectively. For every $t\in \Lambda \setminus B(f_{\Lambda })$ , we have $\deg f_t=d.$ When $\Lambda $ is connected, this is equivalent to say that $\deg f_t=d$ for one $t\in \Lambda \setminus B(f_{\Lambda }).$ A meromorphic family is holomorphic if and only if $B(f_{\Lambda })=\emptyset $ . So, giving a degree d holomorphic family $f_{\Lambda }$ on $\Lambda $ is equivalent to giving a holomorphic morphism $t\mapsto f_t=P_t/Q_t$ from $\Lambda $ to $\text {Rad}_d({\mathbb C}).$ We say that $f_{\Lambda }$ is algebraic if $\Lambda $ is a complex algebraic variety and $f_{\Lambda }: {\mathbb P}^1\times \Lambda \to {\mathbb P}^1\times \Lambda $ is algebraic (i.e., $P_t,Q_t \in {\mathcal O}(\Lambda )[x,y]).$ In other words, it means that the induced morphism $\Lambda \to \text {Rad}_d({\mathbb C})$ is algebraic.

For a degree d holomorphic family $f_{\Lambda }$ on $\Lambda $ , let $\Psi _{\Lambda }: \Lambda \to {\mathcal M}_d$ be the holomorphic morphism sending $t\in \Lambda $ to the class of $f_t$ in ${\mathcal M}_d({\mathbb C}).$ We say that $f_{\Lambda }$ is isotrivial if $\Psi _{\Lambda }: \Lambda \to {\mathcal M}_d$ is locally constant. More generally for degree d meromorphic family $f_{\Lambda }$ , we say that $f_{\Lambda }$ is isotrivial if $f|_{\Lambda \setminus B(f_{\Lambda })}$ is isotrivial.

5.2 Potentially good reduction

Assume that $\Lambda $ is a Riemann surface and $f_{\Lambda }$ is a meromorphic family of degree $d.$

For $b\in \Lambda $ , we say that $f_{\Lambda }$ has potentially good reduction at b if $\Phi _{\Lambda \setminus (B(f_{\Lambda })\cup \{b\})}: \Lambda \to {\mathcal M}_d$ extends to a holomorphic morphism on $(\Lambda \setminus B(f_{\Lambda }))\cup \{b\}.$ In particular, $f_{\Lambda }$ has potentially good reduction at every $b\in \Lambda \setminus B(f_{\Lambda }).$

Lemma 5.1. Assume that $\Lambda $ is an irreducible smooth projective curve. Let $f_{\Lambda }$ be a meromorphic family of degree $d.$ If $f_{\Lambda }$ has potentially good reduction at every point in $\Lambda $ , then $f_{\Lambda }$ is isotrivial.

Proof. Since $f_{\Lambda }$ has potentially good reduction at every point in $B(f_{\Lambda })$ , $\Psi _{\Lambda \setminus B(f_{\Lambda })}: \Lambda \setminus B(f_{\Lambda })\to {\mathcal M}_d$ extends to a holomorphic morphism $\Psi _{\Lambda }:\Lambda \to {\mathcal M}_d.$ Recall that ${\mathcal M}_d({\mathbb C})=\mathrm {Spec \,} ({\mathcal O}(\text {Rat}_d({\mathbb C})))^{\mathrm {PGL}\,_2({\mathbb C})}$ is affine [Sil07, Theorem 4.36(c)]. This follows from the fact that $\text {Rat}_d({\mathbb C})$ is affine and the geometric invariant theory [MF82, Chapter 1]. Since $\Lambda $ is projective, $\Psi _{\Lambda }$ is a constant map. This concludes the proof.

Having potentially good reduction is a local property at b (i.e., for every open neighborhood U of b in $\Lambda $ , $f_{\Lambda }$ has potentially good reduction at b if and only if $f_{U}:=f_{\Lambda }|_{{\mathbb P}^1({\mathbb C})\times U}$ has potentially good reduction at b). Note that there is an open neighborhood U of b which is isomorphic to a disk ${\mathbb D}$ such that $f_{U\setminus \{b\}}$ is holomorphic. So we can focus on the case that $f_{{\mathbb D}}$ is a meromorphic family that is holomorphic on ${\mathbb D}^*.$ We will give another characterization of potentially good reduction via non-Archimedean dynamics.

5.3 Holomorphic family on puncture disk

Let $f_{{\mathbb D}}$ be a a meromorphic family of degree $d\geq 2$ that is holomorphic on ${\mathbb D}^*.$ Let t be the standard coordinate on ${\mathbb D}$ . We can relate $f_{{\mathbb D}}$ to some non-Archimedean dynamics on the field of Laurent’s series ${\mathbb C}((t)).$

Recall that on ${\mathbb C}((t))$ , there is a t-adic norm $|\cdot |$ : Given an element $z=\sum _{n\geq n_0} a_n t^n\neq 0$ , where $n_0\in {\mathbb Z}$ , $a_n\in {\mathbb C}$ and $a_{n_0}\neq 0$ , the t-adic norm of z is $|z|:=e^{-n_0}$ . This norm is non-Archimedean and ${\mathbb C}((t))$ is complete for $|\cdot |$ . Set ${\mathbb L}:=\widehat {\overline {{\mathbb C}((t))}}.$

Write

$$ \begin{align*}f([x:y],t)=([P_t(x,y):Q_t(x,y)], t)\end{align*} $$

where $P_t(x,y), Q_t(x,y)$ are homogenous polynomials of degree d in ${\mathcal O}^{\mathrm {an}}({\mathbb D})[1/t][x,y]$ without common divisors. Since ${\mathcal O}^{\mathrm {an}}({\mathbb D})[1/t]\subseteq {\mathbb C}((t))$ , $f_{{\mathbb D}}$ defines an endomorphism $f_{{\mathbb C}((t))}: [x,y]\mapsto [P_t(x,y):Q_t(x,y)]$ on ${\mathbb P}^1_{{\mathbb C}((t))}$ of degree $d.$ Set $f_{{\mathbb L}}:=f_{{\mathbb C}((t))}\hat {\otimes }_{{\mathbb C}((t))}{\mathbb L}:{\mathbb P}^1_{\mathbb L} \to {\mathbb P}^1_{\mathbb L}$ .

Recall that

(5.1) $$ \begin{align} \overline{{\mathbb C}((t))}=\cup_{n\geq 1}{\mathbb C}((t^{1/n})). \end{align} $$

To get endomorphisms over ${\mathbb C}((t^{1/n}))$ , we introduce some base changes of $f_{{\mathbb D}}$ as follows. Consider the morphism $\phi _n: U_n:={\mathbb D}\to {\mathbb D}$ sending t to $t^n.$ There is $u_n\in {\mathcal O}^{\mathrm {an}}(U_n)$ such that $u_n^n=\phi ^*t.$ Then, $u_n$ is a coordinate on $U_n$ , and we may identify ${\mathbb C}[u_n]$ with ${\mathbb C}[t^{1/n}]$ (hence, we may identify ${\mathbb C}((u_n))$ with ${\mathbb C}((t^{1/n}))$ ). Let $o\in U_n$ be the point defined by $u_n=0.$ The endomorphism on ${\mathbb P}^{1,\mathrm {an}}_{{\mathbb C}((u_n))}$ induced by $f_{U_n}$ is $f_{{\mathbb C}((u_n))}=f_{{\mathbb C}((t))}\hat {\otimes }_{{\mathbb C}((t))}{\mathbb C}((t^{1/n})).$

Lemma 5.2. If $f_{{\mathbb L}}$ has good reduction, then $f_{{\mathbb D}}$ has potentially good reduction at $0$ .

Remark 5.3. The inverse statement of Lemma 5.2 is also true. However, we do not need that direction in this paper. So we leave it to readers.

Proof of Lemma 5.2

By Theorem 4.6, there is $h\in \mathrm {PGL}\,_2({\mathbb L})$ such that ${\mathcal J}(f_{{\mathbb L}})=\{h(x_G)\}.$ Then, $h^{-1}\circ f_{{\mathbb L}}\circ h$ has explicit good reduction. By (5.1) and Remark 4.3, we may assume that $h\in \mathrm {PGL}\,_2({\mathbb C}((t^{1/n})))$ for some $n\geq 1.$ Since ${\mathbb C}(u_n)$ is dense in ${\mathbb C}((u_n))={\mathbb C}((t^{1/n}))$ , by Remark 4.3 again, we may assume that $h\in \mathrm {PGL}\,_2({\mathbb C}(u_n)).$ There is an open neighborhood V of o such that h and $h^{-1}$ are holomorphic on $V\setminus \{o\}$ (i.e., they define holomorphic families $h_{V\setminus \{o\}}$ and $h^{-1}_{V\setminus \{o\}}).$ We may assume further that $V\simeq {\mathbb D}.$ Consider the family $f_V:=h_V^{-1}\circ f_{U_n}|_V\circ h_V$ . Observe that

(5.2) $$ \begin{align} \Psi_{{\mathbb D}^*}\circ \phi|_{V\setminus \{o\}}=\Psi_{V\setminus\{o\}}. \end{align} $$

Then, $f_V$ induces an endomorphism $f_{{\mathbb C}((u))}=f_{{\mathbb C}((t))}\hat {\otimes }_{{\mathbb C}((t))}{\mathbb C}((u))$ on ${\mathbb P}^{1,\mathrm {an}}_{{\mathbb C}((u))}$ , which has good reduction. So $f_V$ is an endomorphism on ${\mathbb P}^1\times V$ . So $\Psi _{V\setminus \{o\}}$ extends to a holomorphic morphism $\Psi _V: V\to {\mathcal M}_d.$ By (5.2), $\Psi _{{\mathbb D}^*}$ is bounded in some neighborhood of $o.$ So $\Psi _{{\mathbb D}^*}$ extends to a holomorphic morphism on ${\mathbb D}$ , which means that $f_{{\mathbb D}}$ has potentially good reduction at $0$ .

The following definition was introduced by Kiwi.

Definition 5.4. [Kiw15] Let $f_{{\mathbb D}}$ be a meromorphic family of degree $d\geq 2$ which is holomorphic on ${\mathbb D}^*.$ We say an endomorphism g is a rescaling limit of $f_{{\mathbb D}}$ (or $f_{{\mathbb D}^*}$ ) (via $(q,M_{{\mathbb D}})$ ) if there is an integer $q\geq 1$ , a finite set $S\subset {\mathbb P}^1({\mathbb C})$ and a meromorphic family $M_{{\mathbb D}}$ of degree 1, such that $M_{{\mathbb D}}$ and $M^{-1}_{{\mathbb D}}$ are holomorphic on ${\mathbb D}^*$ and

$$ \begin{align*} M_t^{-1}\circ f_t^q\circ M_t(z)\to g(z) \end{align*} $$

when $t\to 0$ , uniformly on compact subsets of ${\mathbb P}^1({\mathbb C})\setminus S$ .

The following result was proved by Kiwi.

Proposition 5.5 [Kiw15, Proposition 3.4]

Let $f_{{\mathbb D}}$ be a meromorphic family of degree $d\geq 2$ which is holomorphic on ${\mathbb D}^*$ . Let $M_{{\mathbb D}}$ be a meromorphic family of degree $1$ , such that $M_{{\mathbb D}}$ and $M^{-1}_{{\mathbb D}}$ are holomorphic on ${\mathbb D}^*$ . Then, for all $q\geq 1$ , the following are equivalent:

1. (i) There exist an endomorphism g on ${\mathbb P}^1$ and a finite set $S\subset {\mathbb P}^1({\mathbb C})$ satisfying

   $$ \begin{align*} M_t^{-1}\circ f_t^q\circ M_t(z)\to g(z) \end{align*} $$

   when $t\to 0$ , uniformly on compact subsets of ${\mathbb P}^1({\mathbb C})\setminus S$ .

2. (ii) The point $x=M_{{\mathbb L}}(x_G)$ is fixed by $f_{{\mathbb L}}^q$ and $\widetilde {M_{{\mathbb L}}^{-1}\circ f_{{\mathbb L}}^q\circ M_{{\mathbb L}}}=g.$

In the case where (i) and (ii) hold, $T_x f^q: T_x \to T_x $ can be identified with g after identifying $T_x $ to $T_{x_G}={\mathbb P}^1({\mathbb C})$ via $T_{x_G}M_{{\mathbb L}}: T_{x_G}\to T_{x}$ . Under this identification, S is a finite subset of $T_x$ , which contains all the bad directions of $T_x f^q$ .

Remark 5.6. One may rewrite Definition 5.4 in the following more geometric way. Let $h_{{\mathbb D}}$ be the meromorphic family $h_{\mathbb D}:=M_{\mathbb D}^{-1}\circ f^q_{{\mathbb D}}\circ M_{\mathbb D}$ on ${\mathbb P}^1({\mathbb C})\times {\mathbb D}$ . Then, $h_0=g$ . Moreover, S can be any finite subset containing $S_0$ where $I(h_{{\mathbb D}})=S_0\times \{0\}\subseteq {\mathbb P}^1({\mathbb C})\times {\mathbb D}$ .

Corollary 5.7. Let $x\in {\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}$ be a type 2 fixed point of $f_{{\mathbb L}}.$ Assume that $T_xf_{{\mathbb L}}$ is conjugate to some endomorphism $g: {\mathbb P}^1({\mathbb C})\to {\mathbb P}^1({\mathbb C}).$ Then, there is $n\geq 1$ , such that g is a rescaling limit of $f_{U_n}|_V$ where $f_{U_n}$ is the base change of $f_{{\mathbb D}}$ by the morphism $U_n:={\mathbb D}\to {\mathbb D}$ sending t to $t^n$ as in Section 5.3, and V is an open neighborhood of $o\in U_{n}$ isomorphic to ${\mathbb D}.$

Proof. There is $M_{{\mathbb L}}\in \mathrm {PGL}\,_2({\mathbb L})$ such that $x=M_{{\mathbb L}}(x_G)$ . By (5.1) and Remark 4.3, we may assume that $M_{{\mathbb L}}\in \mathrm {PGL}\,_2({\mathbb C}((t_n^{1/n})))$ for some $n\geq 1.$ Let $f_{U_n}$ be the base change of $f_{{\mathbb D}}$ by the morphism $\phi _n:U_n:={\mathbb D}\to {\mathbb D}$ , sending t to $t^n$ , and pick $u_n$ with $u_n^n=\phi _n^{-1}(t)$ as in Section 5.3. Since ${\mathbb C}(u_n)$ is dense in ${\mathbb C}((u_n))={\mathbb C}((t^{1/n}))$ , by Remark 4.3 again, we may assume that $M_{{\mathbb L}}\in \mathrm {PGL}\,_2({\mathbb C}(u_n)).$ There is an open neighborhood V of o such that $M_{{\mathbb L}}$ and $M_{{\mathbb L}}^{-1}$ are holomorphic on $V\setminus \{o\}$ (i.e., they define holomorphic families $M_{V\setminus \{o\}}$ and $M^{-1}_{V\setminus \{o\}}).$ Then, we conclude the proof by Proposition 5.5.

5.4 Endomorphisms without repelling type I periodic points

In general, the Julia set of an endomorphism $f_{{\mathbb L}}$ on ${\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}$ is a complicated object. The following theorem due to Favre-Rivera Letelier [FRL], and independently by Luo [Luo22, Proposition 11.4], classifies the case when $f_{{\mathbb L}}$ has no repelling type I periodic points.

Theorem 5.8. Let $f_{{\mathbb L}}:{\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}\to {\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}$ be an endomorphism. Assume $f_{{\mathbb L}}$ has no type 1 repelling periodic points. Then, the Julia set of $f_{{\mathbb L}}$ is contained in a segment.

By (v) of Proposition 4.5, ${\mathcal J}(f_{{\mathbb L}})\neq \emptyset .$ In the above theorem, if $f_{{\mathbb L}}$ does not have good reduction, then the segment cannot be a point. As a corollary, we get the following lemma.

Lemma 5.9. Let $f_{{\mathbb L}}:{\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}\to {\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}$ be an endomorphism of degree $d\geq 2$ , which does not have good reduction. Assume that ${\mathcal J}(f_{{\mathbb L}})$ is contained in a minimal segment $[a,b]$ . Let x be a repelling type 2 periodic point in $(a,b)$ with period $q\geq 1$ . Then, the tangent map $T_x f^q$ is conjugate to $z\mapsto z^{m}$ for some $|m|=\deg T_x f^q\geq 2$ . Moreover, every bad direction of $T_x f^q$ is presented by $(x,a]$ or $(x,b]$ and under the above conjugacy, it is identified to $0$ or $\infty $ .

Proof. Since $[a,b]$ is the minimal segment that contains ${\mathcal J}(f)$ , a and b are contained in the Julia set. Since $\deg f_{{\mathbb L}}\geq 2$ and $f_{{\mathbb L}}$ does not have good reduction, the Julia set is not a single point. Hence, $a\neq b$ . Let $v_1$ (resp. $v_2$ ) be the direction in $T_x$ represented by the segment $(x,a]$ (resp. $(x,b]$ ). Since ${\mathcal J}(f_{{\mathbb L}})\subseteq [a,b]$ , $\{v\in T_x|\,\, U(v)\cap {\mathcal J}(f_{{\mathbb L}})\neq \emptyset \}=\{v_1,v_2\}.$ Since ${\mathcal J}(f_{{\mathbb L}})$ is totally invariant, for $v\in T_x$ , if $f^q(U(v))\cap {\mathcal J}(f_{{\mathbb L}})\neq \emptyset $ , then $U(v)\cap {\mathcal J}(f_{{\mathbb L}})\neq \emptyset .$ Hence, $v\in \{v_1,v_2\}.$ This implies $\left \{ v_1,v_2\right \}$ is totally invariant by $T_x f^q$ . Actually, let $w\in (T_x f^q)^{-1}(v_i)$ for some $i=1,2$ . Then, we have $U(v_i)\subset f^q(U(w))$ . This implies $f^q(U(w))\cap {\mathcal J}(f_{{\mathbb L}})\neq \emptyset .$ Thus, $w=v_i$ . Bad directions of $T_x f^q$ are contained in $\left \{ v_1,v_2\right \}$ . Actually, if w is a bad direction, then we have $f^q(U(w))={\mathbb P}_{{\mathbb L}}^{1,\mathrm { an}}$ . Hence, $f^q(U(w))\cap {\mathcal J}(f_{{\mathbb L}})\neq \emptyset $ , which implies $w=v_1$ or $v_2$ . Finally, an endomorphism of degree $\deg T_x f^q$ on ${\mathbb P}^1({\mathbb C})$ has a totally invariant set with two elements that must conjugate to $z\mapsto z^{m}$ for some $|m|=\deg T_x f^q$ . This conjugacy maps $ \left \{ v_1,v_2\right \}$ to $\left \{ 0,\infty \right \}$ , which concludes the proof.

The following theorem is the main result of this section.

Theorem 5.10. Let $f_{{\mathbb D}}$ be a meromorphic family of degree $d\geq 2$ which is holomorphic on ${\mathbb D}^*$ . Assume that $f_{{\mathbb D}}$ does not have potentially good reduction at $0.$ For every $n\geq 1$ , assume that the multipliers of the n-periodic points of $f_t$ are uniformly bounded in t. Then, there is $n\geq 1, m\geq 2$ , such that $g: z\mapsto z^m$ is a rescaling limit of $f_{U_n}|_V$ where $f_{U_n}$ is the base change of $f_{{\mathbb D}}$ by the morphism $U_n:={\mathbb D}\to {\mathbb D}$ , sending t to $t^n$ as in Section 5.3, and V is an open neighborhood of $o\in U_{n}$ isomorphic to ${\mathbb D}.$ Moreover, we may ask the finite set S in Definition 5.4 to be contained in $\left \{ 0,\infty \right \}$ .

Proof. Let $f_{{\mathbb L}}:{\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}\to {\mathbb P}_{{\mathbb L}}^{1,\mathrm {an}}$ be the endomorphism induced by $f_{{\mathbb D}}$ . The multipliers of the n-periodic points of $f_t$ are uniformly bounded in t, which implies $f_{{\mathbb L}}$ has no repelling type 1 periodic points. By Theorem 5.8, ${\mathcal J}(f_{{\mathbb L}})$ is contained in a minimal segment $[a,b]$ . Since $f_{{\mathbb D}}$ does not have potentially good reduction at $0$ , by Lemma 5.2, $f_{{\mathbb L}}$ does not have good reduction. By a result of Rivera-Letelier [BR10, Theorem 10.88], there are infinitely many repelling type 2 periodic points. By (iii) of Proposition 4.5, they are necessarily contained in ${\mathcal J}(f_{{\mathbb L}})$ . Pick a repelling type 2 periodic point x that is contained in $(a,b)$ of period $q\geq 1$ . By Lemma 5.9, replace q by $2q$ if necessary. The tangent map $T_x f^q$ is conjugate to $z\mapsto z^{m}$ for some $m\geq 2$ . Moreover, the bad directions of $T_x f^q$ can be identified with a subset of $\left \{ 0,\infty \right \}$ by the conjugacy. The proof is finished by using Corollary 5.7.

6 A new proof of McMullen’s theorem

We can now give a new proof of Theorem 1.2.

Proof of Theorem 1.2

Let $f_{\Lambda }$ be a non-isotrivial stable irreducible algebraic family of endomorphisms of degree $d\geq 2$ . Since $\Lambda $ is covered by affine open subsets, we may assume that $\Lambda $ itself is affine. Cutting $\Lambda $ by hyperplanes and removing the singular points, we can reduce to the case that $\Lambda $ is a connected Riemann surface of finite type. Since the only non-isotrivial family of exceptional endomorphisms of degree d is the flexible Lattès family, we only need to show that there is a nonempty open subset W of $\Lambda $ such that, for $t\in W$ , $f_t$ , is exceptional.

Write $\Lambda =M\setminus B$ , where M is a compact Riemann surface and B is a finite subset. Since $f_{\Lambda }$ is algebraic, it extends to a meromorphic family of degree $d.$ We have $B(f_M)\subseteq B.$ Since $f_{\Lambda }$ is not isotrivial, by Lemma 5.1, there is $b\in B$ such that $f_M$ does not have potentially good reduction at $b.$ Reparametrize our family near $b\in M$ , and we get a meromorphic family $f_{{\mathbb D}}$ of degree $d\geq 2$ , which is holomorphic on ${\mathbb D}^*$ and preserves the multiplier spectrum.

By Theorem 5.10, after replacing $f_{{\mathbb D}}$ by the family $f_{V}$ in Theorem 5.10, we may assume that $z\mapsto z^m$ for some $m\geq 2$ is a rescaling limit of $f_{{\mathbb D}}$ with $S=\{0,\infty \}.$ Using the reformulation of the rescaling limit in Remark 5.6, there is an integer $q\geq 1$ and a meromorphic family $M_{{\mathbb D}}$ of degree 1, such that $M_{{\mathbb D}}$ and $M^{-1}_{{\mathbb D}}$ are holomorphic on ${\mathbb D}^*$ , and $h_0$ is $z\to z^m$ where $h_{\mathbb D}:=M_{\mathbb D}^{-1}\circ f^q_{{\mathbb D}}\circ M_{\mathbb D}$ on ${\mathbb P}^1({\mathbb C})\times {\mathbb D}$ . Moreover, $I(h_{{\mathbb D}})\subseteq \{0,\infty \}\times \{0\}\subseteq {\mathbb P}^1({\mathbb C})\times {\mathbb D}.$ We may replace $f_{{\mathbb D}}$ by $h_{{\mathbb D}}$ and assume that $f_0: z\mapsto z^m$ and $I(f_{{\mathbb D}})\subseteq \{0,\infty \}\times \{0\}\subseteq {\mathbb P}^1({\mathbb C})\times {\mathbb D}.$

The Julia set of $f_0$ is the unit circle $S^1$ , and $f_0$ is expanding on $S^1$ . We need the following classical lemma of holomorphic motions of expanding sets. A proof can be found (without using quasiconformal maps) in Jonsson [Jon98], which is also valid in higher dimension. Let $K\subset {\mathbb P}^1({\mathbb C})$ be a compact set. We say $f:K\to K$ is expanding if there exist $C>0$ and $\rho>1$ such that $|df^n(x)|\geq C\rho ^n$ for every $n\geq 0$ and $x\in K$ .

Lemma 6.1. Let $(f_t)_{t\in {\mathbb D}}$ be a family of endomorphisms on ${\mathbb P}^1({\mathbb C})$ . Suppose $f_0$ has an expanding set K, $f(K)=K$ . Assume $(f_t)$ is a holomorphic family in a neighborhood of K (i.e., there exists an open set V, $K\subset V$ such that for every $z\in V$ , $t\mapsto f_t(z)$ is holomorphic in ${\mathbb D}$ ). Then, there exist $r>0$ and a continuous map $h:{\mathbb D}_r\times K\to {\mathbb P}^1({\mathbb C})$ such that for each $t\in {\mathbb D}_r$ :

1. (i) $K_t:=h(t,K)$ is an expanding set of $f_t$ .

2. (ii) the map $h_t:=h(t, \cdot ):K\to K_t$ is a homeomorphism and $f_t\circ h_t=h_t\circ f_0$ .

We set $f_0:z\mapsto z^m$ and $K:=S^1$ in the above lemma. The endomorphism $f_0$ has the following properties:

1. (1) $f_0^{-1}(K)=f_0(K)=K;$

2. (2) all periodic points outside the exceptional set $\{0,\infty \}$ are contained in $K;$

3. (3) for every n-periodic point $z\in K$ , we have $df_0^n(z)=m^n$ .

Since the family $(f_t)_{t\in {\mathbb D}^\ast }$ has the same multiplier spectrum, the multiplier of the periodic point $h_t(z)$ of $f_t$ does not change in the family $t\in {\mathbb D}_r^\ast $ . Hence, for every $t\in {\mathbb D}_r$ we have $df_t^n(h_t(z))=m^n$ . We choose a homoclinic orbit $o_i$ , $i\geq 0$ of $f_0$ with $o_0=1$ . By $(1)$ , all $o_i, i\geq 1$ are contained in $K.$ Hence, $h_t(o_i)$ , $i\geq 0$ is a homoclinic orbit of $f_t$ at $z=h_t(1)$ , for $t\in {\mathbb D}_r$ . Let $q_i$ , $i\geq 0$ be an adjoint sequence of $o_i$ , $i\geq 0$ . For every $t\in {\mathbb D}_r^\ast $ , we need to show $h_t(q_i)$ , $i\geq 0$ is an adjoint sequence of $h_t(o_i)$ , $i\geq 0$ . In fact, let $U_t$ be a linearization domain of $f_t$ at $h_t(1)$ . Let $U_{t,i}$ be the connected component of $f_t^{-i}(U_t)$ containing $h_t(o_i)$ . Let l be a good return time of $U_t$ . For every $n\geq l$ , $f_t^n:U_{t,n}\to U_t$ is an isomorphism, with a unique fixed point $p_n$ . Let V be the connected component of $h_t^{-1}(U_t\cap K_t)$ containing $1$ . It is an open arc in $S^1$ . Let $V_n$ be the connected component of $f_0^{-n}(V)$ containing $o_n$ . Since K is totally invariant by $f_0$ and V contains some linearization domain at $1$ , after enlarging l if necessary, for every $n \geq l$ we have $q_n\in V_n\cap K$ . Hence, $h_t(q_n)\in U_{t,n}\cap K_t$ , which is fixed by $f_t^n:U_{t,n}\to U_t$ . By the uniqueness of $p_n$ we have $p_n=h_t(q_n)$ . Hence, $h_t(q_i)$ , $i\geq 0$ is an adjoint sequence of $h_t(o_i)$ , $i\geq 0$ .

For every $t\in {\mathbb D}_r^\ast $ , we consider the dynamics of $f_t$ . The fixed point $h_t(1)$ has multiplier m and the adjoint sequence $h_t(q_i)$ , $i\geq 0$ of the homoclinic orbit $h_t(o_i)$ , $i\geq 0$ has multiplier $m^i$ when i large enough. By Theorem 2.11, $f_t$ is exceptional, which concludes the proof.

7 Conformal expanding repellers and applications

7.1 Definition, examples and rigidity of CER

The following definition was introduced by Sullivan [Sul86].

Definition 7.1. Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ . A compact set $K\subset {\mathbb P}^1({\mathbb C})$ is called a CER of f if

1. (i) there exist $m\geq 1$ and a neighborhood V of K such that $f^m(K)=K$ and $K=\cap _{n\geq 0}f^{-mn}(V)$ .

2. (ii) $f^m:K\to K$ is expanding (i.e., there are constants $C>0$ and $\lambda>1$ such that $|df^{nm}(x)|\geq C\lambda ^n$ for every $x\in K$ and $n\geq 1$ );

3. (iii) $f^m:K\to K$ is topologically exact (i.e., for every open set $U\subset K$ there exists $n\geq 0$ such that $f^{mn}(U)=K$ ).

Remark 7.2. Condition (i)+(ii) is equivalent to $f^m$ expanding on K and $f^m:K\to K$ is an open map [PU10, Lemma 6.1.2].

The following is an important class of examples of CER.

Example 7.3. Assume V, $U_i$ , $1\leq i\leq k$ are connected open sets in ${\mathbb P}^1({\mathbb C})$ , $k\geq 2$ such that $\overline {U_i}\subset V$ , and there exists $m\geq 1$ such that $f^m: U_i\to V$ is an isomorphism. Then, we call

$$ \begin{align*} K:=\left\{ z\in \bigcup_{i=1}^k U_i \Bigg|\,\, f^{mn}(z)\in \bigcup_{i=1}^k U_i\;\text{for every}\;n\geq 0\right\} \end{align*} $$

a horseshoe of f. We check that K satisfies the three conditions in Definition 7.1. Let $V_0:= \cup _{i=1}^k U_i $ .

• Condition (i): It follows from the definition of K;

• Condition (ii): $f^m: V_0 \to V$ strictly expands the hyperbolic metric of V. This implies $f^m:K\to K$ is expanding;

• Condition (iii): Again, using $f^m: V_0 \to V$ strictly expands the hyperbolic metric of V. The maximal diameter of the connected components of $f^{-nm}(V_0)\cap V_0$ shrinks to $0$ when $n\to \infty $ . For each open set $W\subset K$ , there exist integer $n\geq 0$ and a connected component B of $f^{-nm}(V_0)\cap V_0$ such that $B\cap K\subset W$ . Since $f^{(n+1)m}(B\cap K)=K$ , we have $f^{(n+1)m}(W)=K$ . Hence, $f^m:K\to K$ is topologically exact.

Moreover, K is a Cantor set. In particular, K is not a finite set.

When f has degree at least $2$ , there are plenty of horseshoes. Following the terminology in section 2, we can construct a horseshoe associated to finite numbers of homoclinic orbits at o. We prove the following lemma which will be used in the proof of Theorem 1.8.

Lemma 7.4. Let o be a repelling fixed point. Let $k\geq 1$ be an integer. Assume for each fixed $1\leq j\leq k$ , $o_i^{j}$ , $i\geq 0$ is a homoclinic orbit of o such that $o_i^j\notin C(f)$ . Then, there exist an integer $m\geq 1$ and a horseshoe $f^m:K\to K$ such that $o_{mi}^j\in K$ for every $i\geq 0$ and $1\leq j\leq k$ . Moreover, for each $0\leq q\leq m-1$ , $f^q(K)$ is a CER.

Proof. By Lemma 2.6, there exist a linearization domain U of o and an integer m such that, for every $1\leq j\leq k$ , m is a common good return time of U for the homoclinic orbits $o_i^j$ , $i\geq 0$ . Let $U_m^j$ be the connected component of $f^{-m}(U)$ containing $o_m^j$ . Let

$$ \begin{align*} V_0:= \left(\bigcup_{j=1}^k U_m^j\right)\cup g^m(U). \end{align*} $$

Then, the set

$$ \begin{align*} K:=\left\{ z\in V_0 |\,\, f^{mn}(z)\in V_0\;\text{for}\;n\geq 0\right\} \end{align*} $$

is a horseshoe of f. Clearly, we have $o_{mi}^j\in K$ for every $i\geq 0$ and $1\leq j\leq k$ .

For each $0\leq q\leq m-1$ , let $K_q:=f^q(K)$ . We know that $f^q: U_m^j\to U_{m-q}^j$ is an isomorphism, and $f^q:g^m(U)\to g^{m-q}(U)$ is an isomorphism. This implies $f^q:V_0\to f^q(V_0)$ is a finite holomorphic covering (the image of $f^q$ of two components of $V_0$ may coincide). We let $\phi _q$ denote this map. Then we have

$$ \begin{align*} \phi_q\circ f^m|_{V_0}=f^m|_{f^q(V_0)}\circ \phi_q \end{align*} $$

on $f^{-m}(V_0)\cap V_0$ , which implies that $f^m:K\to K$ and $f^m:K_q\to K_q$ are holomorphically semi-conjugated by $\phi _q$ on the corresponding neighborhoods of K and $K_q$ . We check that $K_q$ satisfies the three conditions in Definition 7.1. Since $\phi _q$ is a covering and $f^m:K\to K$ is an open map, $f^m:K_q\to K_q$ is an open map. Since $f^m:K\to K$ is expanding and $|d\phi _q|>c$ on K for some constant $c>0$ , $f^m:K_q\to K_q$ is expanding. By Remark 7.2, conditions (i) and (ii) hold. Since $f^m:K\to K$ is topologically exact and $\phi _q:K\to K_q$ is a semi-conjugacy, $f^m:K_q\to K_q$ is topologically exact. This implies Condition (iii). Hence, $K_q=f^q(K)$ is a CER.

The following definition of linear CER was introduced by Sullivan [Sul86].

Definition 7.5. Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ . Let K be a CER of f. $f(K)=K$ . We call K linear if one of the following conditions holds.

1. (i) The function $\log |df|$ is cohomologous to a locally constant function on K (i.e., there exists a continuous function u on K such that $\log |df|-(u\circ f-u)$ is locally constant on K).

2. (ii) there exists an atlas $\left \{ \phi _i\right \}_{1\leq i\leq k}$ that is a family of holomorphic injections $\phi _i:V_i\to {\mathbb C}$ such that $K\subset \cup _{i=1}^k V_i$ , and all the maps $\phi _i\circ \phi _j^{-1}$ and $\phi _i\circ f\circ \phi _j^{-1}$ are affine.

A proof that these two conditions are actually equivalent can be found in Przytycki-Urbanski [PU10, section 10.1].

The following Sullivan’s rigidity theorem [Sul86] will be used in the proof of Theorem 1.5 and Theorem 1.8. A proof can be found in [PU10, section 10.2].

Theorem 7.6 (Sullivan)

Let $(f,K_f)$ , $(g,K_g)$ be two CERs such that $K_f$ is nonlinear, $f(K_f)=K_f$ , $g(K_g)=K_g$ . Let $h:K_f\to K_g$ be a homeomorphism such that $h\circ f=g\circ h$ on $K_f$ . Then, the following two conditions are equivalent

1. (i) for every periodic point $x\in K_f$ , we have $|df^n(x)|=|dg^n(h(x))|$ , where n is the period of x;

2. (ii) there exist a neighborhood U of $K_f$ and a neighborhood V of $K_g$ such that h extends to a conformal map $h:U\to V$ .

Here, as in Theorem 1.8, a conformal map may change the orientation of ${\mathbb P}^1({\mathbb C})$ .

7.2 Having a linear CER implies exceptional

Now we give a proof of Theorem 1.1.

Proof of Theorem 1.1

Let K be a linear CER of f, which is not a finite set. By [PU10, Proposition 4.3.6], there exists a repelling periodic point $o\in K$ of f. Passing to an iterate of f, we may assume $f(K)=K$ and $f(o)=o$ . Topological exactness of f on K implies for every $a\in K$ , the preimages of $f|_K$ are dense in K. Let U be a linearization domain U of f at o. Since $K\neq \left \{ o\right \}$ , there exist $l\geq 1$ and a point $p_l\in K$ such that $p_l\neq o$ , $f^l(p_l)=o$ . Then, there exists a (unique) homoclinic orbit $o_i$ , $i\geq 0$ such that $o_l=p_l$ and $o_i\in U$ for every $i\geq l$ . Clearly, $o_i\in K$ when $i\leq l$ . By the definition of CER, there exists a neighborhood V of K such that $K=\cap _{n\geq 0}f^{-n}(V)$ . Shrink U if necessary. We assume $U\subset V$ . Hence, for every $i\geq l$ , we have $o_i\in V$ . This implies for every fixed $i\geq 0$ , for every $n\geq 0$ we have $f^n(o_i)\in V$ . Hence, $o_i\in K$ for every $i\geq 0$ .

Let $\left \{V_j\right \}_{1\leq j\leq k}$ be an affine atlas in Definition 7.5. Shrink the linearization domain U, if necessary. We may assume for every $i\geq 0$ , $U_i$ (the connected component of $f^{-i}(U)$ containing $o_i$ ) is contained in some affine chart, say $V_{j(i)}$ . In particular, $U\subset V_{j(0)}$ and $U_i\subset V_{j(0)}$ for every $i\geq l$ . Let $\left \{ q_i\right \}$ , $i\geq 0$ be the adjoint sequence of $o_i$ , $i\geq 0$ . For every large enough integer n, we have $q_n\in U_n$ . For such fixed n, for every $1\leq i\leq n$ , we have $f^{n-i}(q_n)\in U_i\subset V_{j(i)}$ . Let $\lambda _i\in {\mathbb C}^\ast $ be the derivatives of the affine map $\phi _{j(i+1)}\circ f\circ \phi _{j(i)}^{-1}$ , where $0\leq i\leq l-1$ . Let $\lambda \in {\mathbb C}^\ast $ be the derivatives of the affine map $\phi _{j(0)}\circ f\circ \phi _{j(0)}^{-1}$ . Then, we have $df(o)=\lambda $ , and for every n large enough, we have

$$ \begin{align*} df^n(q_n)=\left(\prod_{i=0}^{l-1} \lambda_i \right)\lambda^{n-l}. \end{align*} $$

By Theorem 2.11, f is exceptional. The proof is finished.

7.3 Marked length spectrum rigidity

We now prove Theorem 1.8 by using Theorem 1.1 and Lemma 7.4.

Proof of Theorem 1.8

It is clear that (ii) implies (i). We need to show (i) implies (ii). Assume that h preserves the marked length spectrum on $\Omega $ . If h extends to a global conformal map ${\mathbb P}^1({\mathbb C})\to {\mathbb P}^1({\mathbb C})$ , since $h\circ f=g\circ h$ on ${\mathcal J}(f)$ , the same equality holds on ${\mathbb P}^1({\mathbb C})$ . So we may replace f by its iterate. Passing to an iterate of f, we assume f has a repelling fixed point $o\in \Omega $ and $o\notin PC(f)$ . A result of Eremenko-van Strien [EVS11] says that if a non-Lattès endomorphism f has the property that all the multipliers are real for periodic points contained in a nonempty open set of ${\mathcal J}(f)$ , then ${\mathcal J}(f)$ is contained in a circle. By this result, there are two cases:

1. (i) we can further choose o such that $df(o)\notin {\mathbb R}$ ;

2. (ii) ${\mathcal J}(f)$ is contained in a circle C.

By our choice of o, $h(o)$ is a repelling fixed point of g. Moreover, we have $h(o)\notin PC(g)$ since h preserves critical points in the Julia set. This can be proved using the total invariance of the Julia sets and the fact that critical means locally, not injective. Let $o_i \ i\geq 0$ be a homoclinic orbit of o. Then, $h(o_i)$ , $i\geq 0$ is a homoclinic orbit of $h(o)$ . Let U be a linearization domain of o such that $U\cap {\mathcal J}(f)\subset \Omega $ . Let W be a connected open neighborhood of $h(o)$ such that $h(U\cap {\mathcal J}(f))\subset W$ and $W\cap {\mathcal J}(g)\subset h(\Omega )$ . By Lemma 2.6, shrink U and W, if necessary. There exists $m\geq 1$ such that m is a good return time of U (resp. W) for $o_i, i\geq 0$ (resp. $h(o_i), i\geq 0$ ). By Lemma 7.4, there exist two horseshoes, $f^m:K_f\to K_f$ (resp. $g^m:X_g\to X_g$ ) such that $o_{im}\in K_f$ , $i\geq 0$ (resp. $h(o_{im})\in X_g$ , $i\geq 0$ ). We let $K_g:=h(K_f)$ . By our construction, we have $h:K_f\to K_g$ is a homeomorphism and $h\circ f^m=g^m\circ h$ on $K_f$ . Moreover, $K_g\subset X_g$ . We check that $K_g$ is a CER of g: $g^m:K_g\to K_g$ is open and topologically exact since $f^m:K_f\to K_f$ is; $g^m:K_g\to K_g$ is expanding since $K_g$ is contained in an expanding set $X_g$ . Hence, $K_g$ is a CER of g. Passing to an iterate we may assume $f(K_f)=K_f$ and $g(K_g)=K_g$ . To simplify the notation, for $i\geq 0$ , we let $o_i$ be the unique point in $f^{-i}(o)$ which is contained in the previous homoclinic orbit.

Since f is not exceptional, $K_f$ is a nonlinear CER by Theorem 1.1. Moreover, by our construction, we have $K_f\subset \Omega $ . Hence, for every n-periodic point $x\in K_f$ , we have $|df^n(x)|=|dg^n(h(x))|$ . By Theorem 7.6, h can be extended conformally to a neighborhood V of $K_f$ such that $V\cap {\mathcal J}(f) \subset \Omega $ . We denote this extension by $\tilde {h}$ . In case (ii), we can further assume that $\tilde {h}$ is in fact holomorphic. If $\tilde {h}$ is antiholomorphic on some connected component B of V, let $\phi $ be a nonidentity conformal map (necessarily antiholomorphic) on ${\mathbb P}^1({\mathbb C})$ such that $\phi $ fixes every point in C, then on B. We may replace $\tilde {h}$ by $\tilde {h}\circ \phi $ , which is holomorphic. We have $\tilde {h}=h$ on $K_f$ . Since $\tilde {h}\circ f=g\circ \tilde {h}$ on $K_f$ and $K_f$ is a perfect set, by the conformality of $\tilde {h}$ , we have $\tilde {h}\circ f=g\circ \tilde {h}$ on V.

Next, we show that $\tilde {h}=h$ on $U_0\cap {\mathcal J}(f)$ , where $U_0\subset V$ is a linearization domain of o. Let E be the set of all f-preimages of o. For every $a\in E\cap U_0$ , $f^q(a)=o$ , there exists a homoclinic orbit $o_i'$ of o such that $a=o^{\prime }_q$ , and $o^{\prime }_i\in U_0$ for every $i\geq q$ .

Choose $m'\geq q$ by similar construction as in the first paragraph. We get two CERs, $f^{m'}:K^{\prime \prime }_f\to K^{\prime \prime }_f$ (resp. $g^{m'}:K^{\prime \prime }_g\to K^{\prime \prime }_g$ ) such that $o_{im'}\in K^{\prime \prime }_f$ and $o^{\prime }_{im'}\in K^{\prime \prime }_f$ (resp. $h(o_{im'})\in K^{\prime \prime }_g$ and $h(o^{\prime }_{im'})\in K^{\prime \prime }_g$ ) for $i\geq 0$ . Moreover, $K^{\prime \prime }_f$ is a horseshoe and $K^{\prime \prime }_g$ is contained in a horseshoe $X^{\prime \prime }_g$ . By Lemma 7.4, $K^{\prime }_f:=f^{m'-q}(K^{\prime \prime }_f)$ and $f^{m'-q}(X^{\prime \prime }_f)$ are CERs. Since $K^{\prime }_g:=f^{m'-q}(K^{\prime \prime }_g)\subseteq f^{m'-q}(X^{\prime \prime }_f)$ , $g^m: K^{\prime }_g\to K^{\prime }_g$ is expanding. Since $h:K^{\prime }_f\to K^{\prime }_g$ is a homeomorphism and $h\circ f^{m'}=g^{m'}\circ h$ on $K^{\prime }_f$ , $g^{m'}: K^{\prime }_g\to K^{\prime }_g$ is open and topologically exact. By Remark 7.2, $K^{\prime }_g$ is a CER. Moreover, we have $o_{q+im'}\in K^{\prime }_f$ and $o^{\prime }_{q+im'}\in K^{\prime }_f$ (resp. $h(o_{q+im'})\in K^{\prime }_g$ and $h(o^{\prime }_{q+im'})\in K^{\prime }_g$ ) for $i\geq 0$ . Since f is not exceptional, $K^{\prime }_f$ is a nonlinear CER by Theorem 1.1. Moreover, every periodic point x of $f^{m'}:K^{\prime }_f\to K^{\prime }_f$ has the form $x=f^{m'-q}(y)$ , where y is a periodic point x of $f^{m'}:K^{\prime \prime }_f\to K^{\prime \prime }_f$ . Since $K^{\prime \prime }_f\subset \Omega $ , we get that the f-orbit of x has nonempty intersection with $\Omega $ . This implies for every n-periodic point x of $f^{m'}:K^{\prime }_f\to K^{\prime }_f$ , we have $|df^{m'n}(x)|=|dg^{m'n}(h(x))|$ . By Theorem 7.6, h can be extended conformally to a neighborhood $V'$ of $K^{\prime }_f$ . Denote this extension by $\tilde {h}'$ . In case (ii), we further assume that $\tilde {h}'$ is holomorphic. We have $\tilde {h}'(o_{q+im'})=\tilde {h}(o_{q+im'})=h(o_{q+im'})$ , $i\geq 0$ . The set $\left \{o_{q+im'}, i\geq 0 \right \}$ is a set with accumulation point o. We claim that $\tilde {h}'=\tilde {h}$ on $V_0$ , where $V_0$ is the connected component of $V\cap V'$ containing o. In case (i), since $df(o)\notin {\mathbb R}$ , $\tilde {h}'$ and $\tilde {h}$ are both holomorphic or both antiholomorphic on $V_0$ , hence $\tilde {h}'=\tilde {h}$ on $V_0$ . In case (ii), by our choices $\tilde {h}'$ and $\tilde {h}$ are both holomorphic. Hence $\tilde {h}'=\tilde {h}$ on $V_0$ .

There exists $b\in V_0\cap K^{\prime }_f$ such that $f^{q+nm'}(b)=a$ for some $n\geq 0$ and $\left \{ b, f(b), \cdots , f^{q+nm'}(b)\right \}\subset U_0$ . We also have $\tilde {h}(b)=\tilde {h}'(b)=h(b)$ . Since $\tilde {h}\circ f=g\circ \tilde {h}$ on $U_0$ , we have

$$ \begin{align*} \tilde{h}(a)=\tilde{h}(f^q(b))=g^q(\tilde{h}(b))=g^q(h(b))=h(f^q(b))=h(a). \end{align*} $$

This implies $\tilde {h}=h$ on $E\cap U_0$ . Since E is dense in ${\mathcal J}(f)$ , we get that $\tilde {h}=h$ on $U_0\cap {\mathcal J}(f)$ .

In summary, we have shown that the homeomorphism $h:{\mathcal J}(f)\to {\mathcal J}(g)$ conjugates f to g and can be extended conformally to a disk intersecting ${\mathcal J}(f)$ . By a lemma due to Przytycki-Urbanski [PU99, Proposition 5.4, Lemma 5.5], h extends to a conformal map $h:{\mathbb P}^1({\mathbb C})\to {\mathbb P}^1({\mathbb C})$ such that $h\circ f=g\circ h$ on ${\mathbb P}^1({\mathbb C})$ .

7.4 Marked multiplier spectrum rigidity

Combining Theorem 1.8 and Eremenko-van Strien’s theorem [EVS11], we now prove Theorem 1.7.

Proof of Theorem 1.7

It is clear that (ii) implies (i). We need to show that (i) implies (ii). Assume h preserves the marked multiplier spectrum on $\Omega $ . By Theorem 1.8, h can be extended to a conformal map on ${\mathbb P}^1({\mathbb C})$ . If h is holomorphic, then we are done. If h is antiholomorphic, then the multipliers of all periodic points in $\Omega $ are real. By the main theorem in [EVS11], ${\mathcal J}(f)$ is contained in a circle C. Let $\phi $ be a nonidentity conformal map on ${\mathbb P}^1({\mathbb C})$ such that $\phi $ fixes every point in C. Let $\tilde {h}:=h\circ \phi $ . Then $\tilde {h}\in \mathrm {PGL}\,_2({\mathbb C})$ , and we have $ \tilde {h}\circ f=g\circ \tilde {h}$ on ${\mathbb P}^1({\mathbb C})$ . This finishes the proof.

7.5 Another proof of McMullen’s theorem

Now we can give another proof of Theorem 1.2 using $\lambda $ -Lemma and Theorem 1.7.

Proof of Theorem 1.2

By using $\lambda $ -Lemma [McM16, Theorem 4.1], it is well known that two endomorphisms in a stable family are quasiconformally conjugate on thier Julia sets. Assume by contradiction the conclusion is not true. Since exceptional endomorphisms that are not flexible Lattès are isolated in the moduli space $\mathcal {M}_d$ , there is at least one f in the familly that is not exceptional. Let g be another endomorphism in the family. Let $h:{\mathcal J}(f)\to {\mathcal J}(g)$ be the quasicoformal conjugacy. Since multiplier spectrum is preserved in this family and the conjugacy h moves continuously in the family, for every n-periodic point x of f, we have $df^n(x)=dg^n(h(x))$ . By Theorem 1.7, h extends to an automorphism on ${\mathbb P}^1({\mathbb C})$ . This contradicts the assumption that the family is non-isotrivial.

7.6 Milnor’s conjecture on Lyapunov exponent

We now prove Theorem 1.14 using Theorem 1.1.

Proof of Theorem 1.14

Let S be the finite exceptional set of periodic points in Theorem 1.14. Passing to an iterate of f, there exists a repelling fixed point o of f such that $o\notin S$ . Choose a linearization domain U of o such that $U\cap S=\emptyset $ . By the discussion in Lemma 7.4, there exists a horseshoe $K\subset U$ . Passing to an iterate of f, we assume that $f(K)=K$ . For every n-periodic point $x\in K$ , we have $|df^n(x)|=b^n$ for some $b>0$ . Consider the function $\phi :=\log |df|$ . We have shown that, for every n-periodic point $x\in K$ , $\sum _{i=0}^{n-1}\phi (f^i(x)) =n\log b$ . Recall the following classical Livsic Theorem [Liv72].

Lemma 7.7. Let K be a CER of f, $f(K)=K$ . Let $\phi $ be a Hölder continuous function on K. Assume there exists a constant C such that for every n-periodic point $x\in K$ of f, we have

$$ \begin{align*} \sum_{i=0}^{n-1}\phi(f^i(x))=nC. \end{align*} $$

Then, there exists a continuous function u on K such that $\phi -C=u\circ f-u$ .

Applying the above theorem to $\phi :=\log |df|$ , we get that $\phi $ is cohomologous to a constant function on K in the sense of Definition 7.5. Hence, K is a linear CER, which is not a finite set. By Theorem 1.1, f is exceptional. The proof is finished.

Next, we prove Corollary 1.16. Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ of degree at least $2$ . By Gelfert-Przytycki-Rams [GPR10], there is a forward invariant finite set $\Sigma \subset {\mathcal J}(f)$ with cardinality at most 4 (possibly empty), such that for every finite set $F\subset {\mathcal J}(f)\setminus \Sigma $ , we have $f^{-1}(F)\setminus C(f)\neq F$ . Let $\Delta '(f)$ be the closure of the Lyapunov exponents of periodic points contained in ${\mathcal J}(f)\setminus \Sigma $ . The following theorem was proved by Gelfert-Przytycki-Rams-Rivera Letelier. Be aware that the definition of ‘exceptional’ in [GPR10] and [GPRRL13] has a different meaning.

Theorem 7.8 [GPR10, Theorem 2], [GPRRL13, Theorem 1, Proposition 10]

Let $f:{\mathbb P}^1\to {\mathbb P}^1$ be an endomorphism over ${\mathbb C}$ of degree at least $2$ . Then, $\Delta '(f)$ is a closed interval (possibly a singleton).

Proof of Corollary 1.16

If $\Delta '(f)$ is not a singleton, then we are done by Theorem 7.8. If $\Delta '(f)$ is a singleton, then by Theorem 1.14, f is exceptional, contradicting our assumption. This finishes the proof.

7.7 A simple proof of Zdunik’s theorem

Next, we give a simple proof of Theorem 1.11, using Theorem 1.1.

Proof of Theorem 1.11

It is easy to observe that if f is exceptional, then $\mu $ is absolutely continuous with respect to $\Lambda _{\alpha }$ . We only need to show the converse is true.

Let $\phi :=\alpha \log |df|$ . Following Zdunik [Zdu90], we say $\phi $ is cohomologous to $\log d$ if there exists a function $u\in L^2 ({\mathcal J}(f),\mu )$ such that $\phi -\log d=u\circ f-u$ holds for almost every point, where ${\mathcal J}(f)$ is the Julia set. By a result of Przytycki-Urbanski-Zdunik [PUZ89, Theorem 6], $\phi $ is not cohomologous to $\log d$ , implying $\mu $ is singular with respect to $\Lambda _{\alpha }$ . So we only need to show that $\phi $ is cohomologous to $\log d$ implying f is exceptional. Now we assume $\phi -\log d=u\circ f-u$ for some $u\in L^2 ({\mathcal J}(f),\mu )$ . By a lemma due to Zdunik [Zdu90, Lemma 2], for every $p\notin PC(f)$ , there exists a neighborhood U of p such that u equals to a continuous function almost everywhere. We observe that if $\phi _f:=\alpha \log |df|$ satisfy $\phi _f-\log d=u\circ f-u$ , then $\phi _{f^n}:=\alpha \log |df^n|$ satisfies

(7.1) $$ \begin{align} \phi_{f^n}-n\log d=u\circ f^n-u. \end{align} $$

Passing to an iterate of f, there exists a repelling fixed point $o\notin PC(f)$ . Let U be a linearization domain of o such that u is continous on U. Let K be a horseshoe of f contained in U. Passing to an iterate of f, we may assume $f(K)=K$ . Since u is continuous on K, by (7.1), the function $\log |df|$ is cohomologous to a constant on K in the sense of Definition 7.5. This implies K is a linear CER. Since K is not a finite set, by Theorem 1.1, f is exceptional. The proof is finished.

8 Length spectrum as moduli

For $N\geq 1$ , the symmetric group $S_N$ acts on ${\mathbb C}^N$ (resp. ${\mathbb R}^N$ ) by permuting the coordinates. Using symmetric polynomials, one can show that ${\mathbb C}^N/S_N\simeq {\mathbb C}^N.$ For every element $(\lambda _1,\dots , \lambda _N)\in {\mathbb C}^N$ (resp. ${\mathbb R}^N$ ), we denote by $\{\lambda _1,\dots ,\lambda _N\}$ its image in ${\mathbb C}^N/S_N$ (resp. ${\mathbb R}^N/S_N$ ). We may view the elements in ${\mathbb C}^N/S_N$ as multisets. ⁴

For $d\geq 2,$ let $f_{\text {Rat}_d}: \text {Rat}_d\times {\mathbb P}^1$ be the endomorphism sending $(t,z)$ to $(t, f_t(z))$ where $f_t$ is the endomorphism associated to $t\in \text {Rat}_d$ . For $t\in \text {Rat}_d$ , $f_t^n$ has $N_n:=d^n+1$ fixed points counted with multiplicities. Let $\lambda _1,\dots ,\lambda _{d^n+1}$ be the multipliers of such fixed points. Define $s_n(t)=s_n(f_t):=\{\lambda _1,\dots ,\lambda _{d^n+1}\}\in {\mathbb A}^{N_n}/S_{N_n}$ the n-th multiplier spectrum of $f_t$ . Similarly, define $L_n(t)=L_n(f_t):=\{|\lambda _1|,\dots ,|\lambda _{d^n+1}|\}\in {\mathbb R}^{N_n}/S_{N_n}$ the n-th length spectrum of $f_t$ . Both $s_n(f_t)$ and $L_n(f_t)$ only depend on the conjugacy class of $f_t.$

For every $n\geq 1,$ let $\mathrm {Per}\,_n(f_{\text {Rat}_d})$ be the closed subvariety of $\text {Rat}_d\times {\mathbb P}^1$ of the n-periodic points of $f_{\text {Rat}_d}.$ Let $\phi _n: \mathrm {Per}\,_n(f_{\text {Rat}_d})\to \text {Rat}_d$ be the first projection. It is a finite map of degree $d^n+1.$ Let $\lambda _n: \mathrm {Per}\,_n(f_{\text {Rat}_d})\to {\mathbb A}^1$ be the algebraic morphism $(f_t, x)\mapsto df_t^n(x)\in {\mathbb A}^1$ . Let $|\lambda _n|: \mathrm {Per}\,_n(f_{\text {Rat}_d({\mathbb C})})({\mathbb C})\to [0,+\infty )$ be the composition of $\lambda _n$ to the norm map $z\in {\mathbb C}\mapsto |z|\in [0,+\infty ).$ A fixed point x of $f_t^n$ has multiplicity $>1$ if and only if $df_t^n(x)=1$ . This shows that the map $\phi _n$ is étale at every point $x\in \mathrm {Per}\,_n(f_{\text {Rat}_d})\setminus \lambda _n^{-1}(1).$

We may view $\mathrm {Per}\,_n(f_{\text {Rat}_d})$ as the moduli space of endomorphisms of degree d with a marked n-periodic point. So we may also denote it by $\text {Rat}_d[n]$ or $\text {Rat}^1_d[n]$ . More generally, for every $s=1,\dots , d^n+1$ , one may construct the moduli space $\text {Rat}^s_d[n]$ of endomorphisms of degree d with s marked n-periodic point as follows: For $s=2,\dots , d^n+1,$ consider the fiber product $(\text {Rat}_d[n])^s_{/\text {Rat}_d}$ of s copies of $\text {Rat}_d[n]$ over $\text {Rat}_d.$ For $i\neq j\in \{1,\dots , d^n+1\}$ , let $\pi _{i,j}: (\text {Rat}_d[n])^s_{/\text {Rat}_d}\to (\text {Rat}_d[n])^2_{/\text {Rat}_d}$ be the projection to the $i,j$ coordinates. The diagonal $\Delta \subseteq (\text {Rat}_d[n])^2_{/\text {Rat}_d}$ is an irreducible component of $(\text {Rat}_d[n])^2_{/\text {Rat}_d}$ . One may define $\text {Rat}^s_d[n]$ to be the Zariski closure of

$$ \begin{align*}(\text{Rat}_d[n])^s_{/\text{Rat}_d}\setminus (\cup_{i\neq j\in \{1,\dots, d^n+1\}}\pi_{i,j}^{-1}(\Delta))\end{align*} $$

in $(\text {Rat}_d[n])^s_{/\text {Rat}_d}.$ Denote by $\phi _n^s: \text {Rat}^s_d[n]\to \text {Rat}_d$ the morphism induced by $\phi _n$ . Let $\lambda ^s_n: \text {Rat}^s_d[n]\to {\mathbb A}^s$ be the morphism defined by $(t, x_1,\dots ,x_s)\mapsto (df^n(x_1),\dots , df^n(x_s))$ and $|\lambda ^s_n|: \text {Rat}^s_d[n]({\mathbb C})\to {\mathbb R}^s$ the map defined by $(t, x_1,\dots ,x_s)\mapsto (|df^n(x_1)|,\dots , |df^n(x_s)|).$ Since $\phi _n$ is étale at every point $x\in \mathrm {Per}\,_n(f_{\text {Rat}_d})\setminus \lambda _n^{-1}(1),$ $\phi ^s_n$ is étale at every point $x\in (\lambda _n^s)^{-1}(({\mathbb A}^1\setminus \{1\})^s).$

To prove Theorem 1.5, we need to study the subsets taking the form $\Lambda _n(a):=L_n^{-1}(a)$ where $a\in {\mathbb R}^{N_n}/S_{N_n}.$ Since $L_n$ is not holomorphic (hence, not algebraic), in general, the above set is not algebraic. The problem is that one projects a real algebraic set under a finite map, but it may not be real algebraic. To get some algebricity of $\Lambda _n(a)$ , one can view $\text {Rat}_d({\mathbb C})$ as a real algebraic variety by splitting a complex variable z into two real varieties $x,y$ via $z=x+iy$ . A more theoretic way to do this is using the notion of Weil restriction. See Section 8.1.1 for a brief introduction. However, even when we view $\text {Rat}_d({\mathbb C})$ as a real algebraic variety, $\Lambda _n(a)$ is not real algebraic in general (c.f. Theorem 8.10). Here, real algebraic means Zariski closed when viewing $\text {Rat}_d({\mathbb C})$ as a real algebraic variety. See Section 8.1.1 for the precise definition. This is one of the main difficulties in the proof of Theorem 1.5. To solve this problem, we introduce a class of closed subsets of $\text {Rat}_d({\mathbb C})$ that are images of algebraic subsets under étale morphisms. We will study such subsets in Section 8.2.

8.1 An example of a length level set which is not real algebraic

The main result of this section is Theorem 8.10, in which we give an example to show that the subsets $\Lambda _n(a)$ may not be real algebraic in $\text {Rat}_d({\mathbb C})$ ⁵.

Except Definition 8.1, in which we give a precise definition of the notion real algebraic using Weil restriction, this section will not be used in the rest of the paper.

8.1.1 Weil restriction

We briefly recall the notion of Weil restriction. See [Poo17, Section 4.6] and [BLR90, Section 7.6] for more information.

Denote by $Var_{/{\mathbb C}}$ (resp. $Var_{/{\mathbb R}}$ ) the category of varieties over ${\mathbb C}$ (resp. ${\mathbb R}$ ). For every variety X over ${\mathbb C}$ , there is a unique variety $R(X)$ over ${\mathbb R}$ representing the functor $Var_{/{\mathbb R}}\to Sets$ , sending $V\in Var_{/{\mathbb R}}$ to $\mathrm {Hom}(V\otimes _{{\mathbb R}}{\mathbb C}, X).$ It is called the Weil restriction of X. The functor $X\mapsto R(X)$ is called the Weil restriction. One has the canonical morphism $\tau _X: X({\mathbb C})\to R(X)({\mathbb R}),$ which is a real analytic diffeomorphism. One may view $X({\mathbb C})$ as a real algebraic variety via $\tau _X.$

Definition 8.1. The real Zariski topology on $X({\mathbb C})$ is the restriction of the Zariski topology on $R(X)$ via $\tau _X$ . A subset Y of $X({\mathbb C})$ is real algebraic if it is closed in the real Zariski topology.

By (iii) of Proposition 8.3 below, the real Zariski topology is stronger than the Zariski topology on $X({\mathbb C}).$

Roughly speaking, the Weil restriction is just constructed by splitting a complex variable z into two real variables $x,y$ via $z=x+iy$ . For the convenience of the reader, in the following example, we show the concrete construction of $R(X)$ when X is affine.

Example 8.2. First assume that $X={\mathbb A}^N_{{\mathbb C}}$ . Then $R(X)={\mathbb A}^{2N}_{{\mathbb R}}.$ The map

$$ \begin{align*}\tau_{X}: {\mathbb A}^N_{{\mathbb C}}({\mathbb C})={\mathbb C}^N\to {\mathbb A}^{2N}_{{\mathbb R}}({\mathbb R})={\mathbb R}^{2N}\end{align*} $$

sends $(z_1, \dots , z_N)$ to $(x_1, y_1, x_2,y_2,\dots , x_N,y_N)$ where $z_j=x_j+iy_j$ .

Consider the algebra ${\mathbb B}:={\mathbb C}[I]/(I^2+1)\simeq {\mathbb C}\oplus I{\mathbb C}$ . Every $f\in {\mathbb C}[z_1,\dots ,z_N]$ defines an element

$$ \begin{align*}F:=f(x_1+Iy_1, \dots, x_N+Iy_N)\in {\mathbb B}[x_1,y_1,\dots, x_N,y_N].\end{align*} $$

Since

$$ \begin{align*}{\mathbb B}[x_1,y_1,\dots, x_N,y_N]={\mathbb C}[x_1,y_1,\dots, x_N,y_N]\oplus I{\mathbb C}[x_1,y_1,\dots, x_N,y_N],\end{align*} $$

F can be uniquely decomposed to $F=r(f)+Ii(f)$ where $r(f), i(f)\in {\mathbb C}[x_1,y_1,\dots ,x_N,y_N].$

If X is the closed subvariety of ${\mathbb A}^N_{\mathbb C}=\mathrm {Spec \,}{\mathbb C}[z_1,\dots , z_M]$ defined by the ideal $(f_1,\dots , f_s)$ , then $R(X)$ is the closed subvariety of $R({\mathbb A}^N_{{\mathbb C}})={\mathbb A}^{2N}_{{\mathbb R}}=\mathrm {Spec \,} {\mathbb R}[x_1,y_1,\dots , x_N,y_N]$ defined by the ideal generated by $r(f_1), i(f_1),\dots , r(f_s), i(f_s)$ .

We list some basic properties of the Weil restriction without proof.

Proposition 8.3. Let $X, Y\in Var_{/{\mathbb C}}$ . Then, we have the following properties:

1. (i) if X is irreducible, then $R(X)$ is irreducible;

2. (ii) $\dim R(X)=2\dim X;$

3. (iii) if $f:Y\to X$ is a closed (resp. open) immersion, then the induced morphism $R(f):R(Y)\to R(X)$ is a closed (resp. open) immersion.

Then, we get the following easy consequence.

Lemma 8.4. Let $Y\in Var_{{\mathbb C}}$ and X be a closed subset Y. Then, $R(X)$ is the Zariski closure of $X({\mathbb C})=R(X)({\mathbb R})$ in $R(Y).$

Proof. We may assume that X and Y are irreducible. It is clear that $R(X)({\mathbb R})\subseteq R(X)$ . So $\overline {R(X)({\mathbb R})}^{\mathrm { zar}}\subseteq R(X).$ Since

$$ \begin{align*}\dim_{{\mathbb R}}\overline{R(X)({\mathbb R})}^{\mathrm{zar}}\geq \dim_{{\mathbb R}} R(X)({\mathbb R})=2\dim X=\dim R(X)\end{align*} $$

and $R(X)$ is irreducible, we get $\overline {R(X)({\mathbb R})}^{\mathrm {zar}}= R(X).$

We denote by $\sigma \in \mathrm {Gal}({\mathbb C}/{\mathbb R})$ the complex conjugation $z\mapsto \overline {z}$ . For every complex variety X, one denotes by $X^{\sigma }$ the base change of X by the field extension $\sigma : {\mathbb C}\to {\mathbb C}$ . This induces a morphism of schemes (over ${\mathbb Z}$ ) $\sigma : X^{\sigma }\to X$ . It is not a morphism of schemes over ${\mathbb C}$ . It is clear that $(X^{\sigma })^{\sigma }=X.$

Example 8.5. If X is the subvariety of ${\mathbb A}^N_{{\mathbb C}}=\mathrm {Spec \,} {\mathbb C}[z_1,\dots , z_N]$ defined by the equations $\sum _{I}a_{i,I}z^I=0, i=1,\dots , s$ , then $X^{\sigma }$ is the subvariety of ${\mathbb A}^N_{{\mathbb C}}$ defined by $\sum _{I}\overline {a_{i,I}}z^I=0, i=1,\dots , s$ . The map $\sigma : X=(X^{\sigma })^{\sigma }\to X^{\sigma }$ sends a point $(z_1,\dots ,z_N)\in X({\mathbb C})$ to $(\overline {z_1},\dots ,\overline {z_N})\in X^{\sigma }({\mathbb C})$ .

The following result due to Weil is useful for computing the Weil restriction.

Proposition 8.6 [Poo17, Exercise 4.7]

We have a canonical isomorphism

$$ \begin{align*}R(X)\otimes_{{\mathbb R}}{\mathbb C}\simeq X\times X^{\sigma}.\end{align*} $$

Under this isomorphism,

$$ \begin{align*}R(X)({\mathbb R})=\{(z_1,z_2)\in X({\mathbb C})\times X^{\sigma}({\mathbb C})|\,\, z_2=\sigma(z_1)\}\end{align*} $$

and $\tau _X$ sends $z\in X({\mathbb C})$ to $(z,\sigma (z))\in R(X)({\mathbb R}).$

8.1.2 The norm map

For $N\geq 1$ , let $\nu _N: {\mathbb C}^N/S_N\to {\mathbb R}^N/S_N$ be the real analytic map sending $\{z_1,\dots ,z_N\}$ to $\{|z_1|^2,\dots , |z_N|^2\}.$ We view ${\mathbb C}^N/S_N$ as a real algebraic variety via the identification

$$ \begin{align*}{\mathbb C}^N/S_N=({\mathbb A}^N_{{\mathbb C}}/S_N)({\mathbb C})=R({\mathbb A}^N_{{\mathbb C}}/S_N)({\mathbb R})\subseteq R({\mathbb A}^N_{{\mathbb C}}/S_N)({\mathbb C}).\end{align*} $$

The following result is the aim of this section. We postpone its proof to the end of this section.

Proposition 8.7. For $a:=\{a_1,\dots ,a_N\}\in {\mathbb R}_{>0}^N/S_N$ , $\nu _N^{-1}(a)$ is real Zariski closed if and only if $N=1$ or $N=2$ and $a_1\neq a_2.$

Set $X:=R({\mathbb A}^N_{{\mathbb C}}/S_N)\otimes _{\mathbb R}{\mathbb C}=({\mathbb A}^N_{{\mathbb C}}/S_N)\times ({\mathbb A}^N_{{\mathbb C}}/S_N).$ (Since ${\mathbb A}^N_{{\mathbb C}}/S_N$ is defined over ${\mathbb R}$ , we have ${\mathbb A}^N_{{\mathbb C}}/S_N=({\mathbb A}^N_{{\mathbb C}}/S_N)^\sigma $ .) Consider the quotient morphisms $q_1: {\mathbb A}_{{\mathbb C}}^N\twoheadrightarrow {\mathbb A}_{{\mathbb C}}^N/S_N$ defined by

$$ \begin{align*}(z_1,\dots, z_N)\mapsto \{z_1,\dots, z_N\}\end{align*} $$

and $q_2: {\mathbb A}_{{\mathbb C}}^N\times {\mathbb A}_{{\mathbb C}}^N\twoheadrightarrow X$ defined by

$$ \begin{align*}(u_1,\dots, u_N; v_1,\dots, v_N)\mapsto (\{u_1,\dots, u_N\},\{v_1,\dots, v_N\}).\end{align*} $$

Consider the morphism $\mu _N: {\mathbb A}_{{\mathbb C}}^N\times {\mathbb A}_{{\mathbb C}}^N\to {\mathbb A}_{{\mathbb C}}^N$ defined by

$$ \begin{align*}(u_1,\dots, u_N; v_1,\dots, v_N)\mapsto (u_1v_1,\dots, u_Nv_N).\end{align*} $$

Let $\Gamma _{\mu _N}$ be the graph of $\mu _N$ in $({\mathbb A}_{{\mathbb C}}^N\times {\mathbb A}_{{\mathbb C}}^N)\times {\mathbb A}_{{\mathbb C}}^N$ . Set $\Gamma _N=(q_2\times q_1)(\Gamma _{\mu _N})\subseteq X\times ({\mathbb A}^N_{{\mathbb C}}/S_N)$ . Since $q_2\times q_1$ is finite, $\Gamma _N$ is an irreducible closed subvariety of $X\times ({\mathbb A}^N_{{\mathbb C}}/S_N)$ . We view it as a correspondence between X and ${\mathbb A}^N_{{\mathbb C}}/S_N$ .

Let $\pi _1: X\times ({\mathbb A}^N_{{\mathbb C}}/S_N)\to X$ and $\pi _2: X\times ({\mathbb A}^N_{{\mathbb C}}/S_N)\to ({\mathbb A}^N_{{\mathbb C}}/S_N)$ be the first and the second projection. Then, $\pi _1|_{\Gamma _N}$ is a finite morphism of degree $N!.$ For every $x\in X$ , the image of x under $\Gamma _N$ is $\Gamma _N(x):=\pi _2(\Gamma _N\cap \pi _1^{-1}(x))$ . For a general $x\in X({\mathbb C})$ , $\Gamma _N(x)$ has $N!$ points. Similarly, for every $y\in {\mathbb A}^N_{{\mathbb C}}/S_N$ , the preimage of y under $\Gamma _N$ is $\Gamma _N^{-1}(y):=\pi _1(\Gamma _N\cap \pi _2^{-1}(y)).$

Lemma 8.8. For every $a=\{a_1,\dots , a_N\}\in ({\mathbb A}^N_{{\mathbb C}}/S_N)({\mathbb C})$ with $a_i\neq 0, i=1,\dots , N$ , $\Gamma _N^{-1}(a)$ is irreducible and of dimension N.

Proof. Consider the actions of $g\in S_N$ on ${\mathbb A}_{{\mathbb C}}^N\times {\mathbb A}_{{\mathbb C}}^N$ by

$$ \begin{align*}g.(u_1,\dots, u_N; v_1,\dots, v_N)=(u_{g(1)},\dots, u_{g(N)}; v_{g(1)},\dots, v_{g(N)})\end{align*} $$

and on ${\mathbb A}^N_{{\mathbb C}}$ by $g.(z_1,\dots , z_N)=(z_{g(1)},\dots , z_{g(N)}).$ Then, we have

$$ \begin{align*}q_1(g.x)=q_1(x), q_2(g.x)=q_2(x).\end{align*} $$

Since

$$ \begin{align*}\Gamma_N^{-1}(a)=q_2(\mu_N^{-1}(q_1^{-1}(\{a_1,\dots, a_N\})))\end{align*} $$

and

$$ \begin{align*}q_1^{-1}(\{a_1,\dots, a_N\})=\{g. (a_1,\dots, a_N)|\,\, g\in S_N\},\end{align*} $$

we get $\Gamma _N^{-1}(a)=q_2(\mu _N^{-1}((a_1,\dots ,a_N))).$ Since $\mu _N^{-1}((a_1,\dots ,a_N))$ is defined by $u_iv_i=a_i, i=1,\dots , N$ , it is isomorphic to $({\mathbb A}^1\setminus \{0\})^N$ , which is irreducible. Since $q_2$ is finite, $\Gamma _N^{-1}(a)$ is irreducible of dimension $N.$

For $a=\{a_1,\dots , a_N\}\in {\mathbb R}_{>0}^N/S_N\subseteq ({\mathbb A}^N_{{\mathbb C}}/S_N)({\mathbb R})$ , we have

$$ \begin{align*}\Gamma_N^{-1}(a)({\mathbb R})=\Gamma_N^{-1}(a)\cap X({\mathbb R})=\cup_{g\in S_N}V_{N,g}(a)\end{align*} $$

where

$$ \begin{align*}V_{N,g}(a)=q_2(\{(u_1,\dots, u_N;\overline{u_1},\dots, \overline{u_N})\in {\mathbb C}^{2N}|\,\, u_i\overline{u_{g(i)}}=a_i,1\leq i\leq N\})\end{align*} $$

$$ \begin{align*}=\{(\{u_1,\dots, u_N\},\{\overline{u_1},\dots, \overline{u_N}\})\in R(X)({\mathbb R})|\,\, u_i\overline{u_{g(i)}}=a_i, 1\leq i\leq N\}\end{align*} $$

We note that, if $g_1,g_2\in S_N$ are conjugate, then $V_{N,g_1}(a)=V_{N,g_2}(a)$ . For every $g\in S_N$ , it can be uniquely written as a product of disjoint cycles (i.e., there is a partition $\{1,\dots ,N\}=\sqcup _{i=1}^s I_i$ such that $g=\sigma _1\cdots \sigma _s$ where $\sigma _i$ acts trivially outside $I_i$ and transitively on $I_i).$ Set

$$ \begin{align*}Z_{N,g}(a):=\{(u_1,\dots, u_N;\overline{u_1},\dots, \overline{u_N})\in {\mathbb C}^{2N}|\,\, u_i\overline{u_{g(i)}}=a_i, i=1,\dots, N\}.\end{align*} $$

Then, $V_{N,g}(a)=q_2(Z_{N,g}(a)).$

For $i=1,\dots ,s$ , set $m_i:=\#I_i$ and write $I_i=\{j_1,\dots , j_{m_i}\}$ with $\sigma (j_n)=j_{n+1}$ . Here, the index n is viewed in ${\mathbb Z}/m_i{\mathbb Z}.$ We define $Z_i, i=1,\dots , s$ as follows:

• $(E_0):$ If $m_i$ is even and $\sum _{n=1}^{m_i}(-1)^n\log a_{j_n}\neq 0$ , $Z_i:=\emptyset $ .

• $(E_1):$ If $m_i$ is even and $\sum _{n=1}^{m_i}(-1)^n\log a_{j_n}= 0,$ then $Z_i$ is the set of points taking forms $(U,\overline {U})\in {\mathbb C}^{I_i}\times {\mathbb C}^{I_i}$ where

  $$ \begin{align*}U=(r_{j_1}e^{i\theta},a_1r^{-1}_{j_1}e^{i\theta}, a_2a_1^{-1}r_{j_1}e^{i\theta}, \dots, a_{j_{m_i-1}}a_{j_{m_i-2}}^{-1}\dots a_1r_{j_i}^{-1}e^{i\theta})\end{align*} $$
  for some $r_{j_1}\in {\mathbb R}_{>0}$ and $\theta \in {\mathbb R}.$ Hence, $Z_i\simeq {\mathbb R}_{> 0}\times ({\mathbb R}/{\mathbb Z}).$

• $(O):$ If $m_i$ is odd, then $Z_i$ is the set of points taking forms $(U,\overline {U})\in {\mathbb C}^{I_i}\times {\mathbb C}^{I_i}$ where

  $$ \begin{align*} U=(r_{j_1}e^{i\theta},\dots, r_{j_{m_i}}e^{i\theta}),r_{j_n}=\left(\prod_{l=0}^{m_i-1}a_{j_{n+l}}^{(-1)^l}\right)^{1/2} \end{align*} $$
  for some $\theta \in {\mathbb R}.$ Hence, $Z_i\simeq {\mathbb R}/{\mathbb Z}.$

It is easy to show that

$$ \begin{align*}Z_{N,g}(a)=\prod_{i=1}^sZ_i.\end{align*} $$

Let $e_0(g), e_1(g)$ and $o(g)$ be the numbers of the index i that falls into the cases $(E_0), (E_1)$ and $(O)$ respectively. Then, $Z_{N,g}(a)=\emptyset $ if $e_0(g)>0$ . Otherwise,

$$ \begin{align*}Z_{N,g}(a)\simeq {\mathbb R}_{>0}^{e_1(g)}\times ({\mathbb R}/{\mathbb Z})^{e_1(g)+o(g)}.\end{align*} $$

Lemma 8.9. We have $V_{N,\mathrm {id}}(a)=\nu _N^{-1}(a)$ , and it is Zariski dense in $\Gamma _N^{-1}(a).$

Proof. It is clear that $V_{N,\mathrm {id}}(a)=\nu _N^{-1}(a)$ . By Lemma 8.8, $\Gamma _N^{-1}(a)$ is irreducible and of dimension N. Since $Z_{N,\mathrm {id}}(a)\simeq ({\mathbb R}/{\mathbb Z})^N$ , $V_{N,\mathrm {id}}(a)=q_2(Z_{N,\mathrm {id}}(a))$ is of dimension N. Then, it is Zariski dense in $\Gamma _N^{-1}(a).$

Proof of Proposition 8.7

By Lemma 8.9, $\nu _N^{-1}(a)=V_{N,\mathrm {id}}(a)$ is Zariski closed if and only if $V_{N,g}(a)\subseteq V_{N,\mathrm {id}}(a)$ for every $g\in S_N.$

The case $N=1$ is trivial. If $N=2$ and $a_1\neq a_2$ , then $e_0(g)>0$ for $g\in S_2\setminus \{\mathrm {id}\}$ . Hence, $V_{N,\mathrm {id}}(a)$ is Zariski closed. If there is $i\neq j$ with $a_i=a_j$ , let $g:=(i,j)\in S^N.$ Then

$$ \begin{align*}Z_{N,g}(a)\simeq {\mathbb R}_{>0}\times ({\mathbb R}/{\mathbb Z})^{N-1}\end{align*} $$

which is not compact. Since $q_2$ is finite, $q_2(Z_{N,g}(a))$ is closed but not compact. Hence, it is not contained in $V_{N,\mathrm {id}}(a)$ .

Now we may assume that $N\geq 3$ and $a_i\neq a_j$ for every $i\neq j$ . We may assume that $a_1>a_2>a_3$ and $a_1=\max \{a_i, i=1,\dots , N\}.$ Then, for every $(\{u_1,\dots , u_N\}, \{\overline {u_1},\dots , \overline {u_N}\})\in V_{N,\mathrm {id}}(a)$ , we have

$$ \begin{align*}\max\{|u_i|, i=1,\dots, N\}=a_1^{1/2}.\end{align*} $$

Pick $g=(1,2,3)\in S_N.$ Then $Z_{N,\mathrm {id}}(a)\neq \emptyset $ and for every point $(u_1,\dots , u_N;\overline {u_1},\dots , \overline {u_N})\in Z_{N,g}(a)$ , we have

$$ \begin{align*}\max\{|u_i|, i=1,\dots, N\}\geq |u_2|=(a_1a_2a_3^{-1})^{1/2}>a_1^{1/2}.\end{align*} $$

Since $V_{N,\mathrm {id}}(a)=q_2(Z_{N,\mathrm {id}}(a))$ , $V_{N,g}(a)\cap V_{N,\mathrm {id}}(a)=\emptyset .$ Hence, $V_{N,\mathrm {id}}(a)$ is not Zariski closed.

8.1.3 The example

In this section, we focus on the first length spectrum map $L_1: \mathrm {Rat}_2({\mathbb C})\to {\mathbb R}_{>0}^3/S_3.$ We view $\mathrm {Rat}_2({\mathbb C})$ as a real algebraic variety via identifying $\mathrm {Rat}_2({\mathbb C})$ with $R(\mathrm {Rat}_2)({\mathbb R})$

Theorem 8.10. For $a\in (1,\sqrt {2})$ , $L_1^{-1}(\{a,a,a\})$ is not real algebraic in $\mathrm {Rat}_2({\mathbb C}).$

Proof. We follow the notations in Section 8.1.2.

Recall the first multiplier spectrum map $s_1: \mathrm {Rat}_2({\mathbb C})\to ({\mathbb A}^3/S_3)({\mathbb C}).$ Then, $L_1^{-1}(\{a,a,a\})=s_1^{-1}(\nu _3^{-1}(\{a^2,a^2,a^2\})).$ Set $b:=\{a^2,a^2,a^2\}.$ Since $s_1$ factors through the moduli space ${\mathcal M}_2({\mathbb C})$ , there is a morphism $[s_1]: {\mathcal M}_2({\mathbb C})\to ({\mathbb A}^3/S_3)({\mathbb C})$ such that $[s_1]\circ \Psi _2=s_1.$ It was proved by Milnor[Mil93] that $[s_1]$ is an isomorphism to its image M (see also [Sil12, Theorem 2.4.5]). Moreover, by [Sil12, Theorem 2.4.5 and Lemma 2.4.6], $M=q_1(Y_0)$ and $R(M)=q_2(R(Y_0))$ , where

$$ \begin{align*}Y_0=\{(z_1,z_2,z_3)\in {\mathbb C}^3|\,\, z_1z_2z_3=z_1+z_2+z_3-2, z_1z_2\neq 1\}\cup \{(1,1,z_3)\}.\end{align*} $$

Set $Y:=\{(z_1,z_2,z_3)\in {\mathbb C}^3|\,\, z_1z_2z_3=z_1+z_2+z_3-2\}$ , which is the Zariski closure of $Y_0.$ The Zariski closure of $R(M)$ in $R({\mathbb A}^3_{{\mathbb C}}/S_3)$ is $q_2(R(Y))$ .

Lemma 8.11. The intersection $q_2(R(Y))\cap \Gamma ^{-1}_3(b)$ is irreducible of dimension $1.$

Proof. Observe that $(q_2(R(Y))\cap \Gamma ^{-1}_3(b))\otimes _{{\mathbb R}}{\mathbb C}=q_2(Z)$ where Z is the closed subset of $R({\mathbb A}^3_{{\mathbb C}})\otimes _{{\mathbb R}} {\mathbb C}={\mathbb A}_{{\mathbb C}}^{3}\times {\mathbb A}_{{\mathbb C}}^{3}=\mathrm {Spec \,} {\mathbb C}[u_1,u_2,u_3,v_1,v_2,v_3]$ defined by the following equations:

1. (i) $u_1u_2u_3=u_1+u_2+u_3-2;$

2. (ii) $v_1v_2v_3=v_1+v_2+v_3-2;$

3. (iii) $u_1v_1=a$ ;

4. (iv) $u_2v_2=a$ ;

5. (v) $u_3v_3=a$ .

Using symmetric polynomials, one may write

$$ \begin{align*}R({\mathbb A}_{{\mathbb C}}^3/S_3)\otimes_{{\mathbb R}}{\mathbb C}={\mathbb A}_{{\mathbb C}}^3/S_3\times {\mathbb A}_{{\mathbb C}}^3/S_3\end{align*} $$

as

$$ \begin{align*}{\mathbb A}_{{\mathbb C}}^3\times {\mathbb A}_{{\mathbb C}}^3=\mathrm{Spec \,} {\mathbb C}[x,y,z,x',y',z']\end{align*} $$

and in this coordinate, $q_2$ is given by $x\mapsto u_1+u_2+u_3$ , $y\mapsto u_1u_2+u_1u_3+u_2u_3$ , $z\mapsto u_1u_2u_3$ , $x'\mapsto v_1+v_2+v_3$ , $y'\mapsto v_1v_2+v_1v_3+v_2v_3$ and $z'\mapsto v_1v_2v_3.$ One may compute that $q_2(Z)$ is defined by the following equations:

1. (i) $z\neq 0;$

2. (ii) $x=z+2;$

3. (iii) $y=(2z+a^3)/a;$

4. (iv) $x'=a^3/z+2$ ;

5. (v) $y'=a^2(z+2)/z$ ;

6. (vi) $z'=a^3/z$ .

Then, it is irreducible of dimension $1$ since it is parametrized by a single variable z.

Then, $R(M)\cap \Gamma ^{-1}_3(b)$ is irreducible, and if this intersection is nonempty, it is of dimension $1.$ We note that

$$ \begin{align*}\nu_3^{-1}(b)=M\cap q_2(Z_{3,\mathrm{id}}(b)).\end{align*} $$

Let $g=(1,2)\in S_3$ . We have

$$ \begin{align*}M\cap (q_2(Z_{3,\mathrm{id}}(b))\cup q_2(Z_{3,g}(b)))\subseteq (R(M)\cap \Gamma^{-1}_3(b))({\mathbb R}).\end{align*} $$

Lemma 8.12. Both $M\cap q_2(Z_{3,\mathrm {id}}(b))$ and $M\cap q_2(Z_{3,g}(b))$ are infinite and $M\cap q_2(Z_{3,g}(b))\not \subseteq M\cap q_2(Z_{3,\mathrm { id}}(b)).$

Proof. Since $q_2$ is finite, we only need to show that $Y_0\cap Z_{3,\mathrm {id}}(b)$ and $Y\cap Z_{3,g}(b)$ are infinite and $M\cap q_2(Z_{3,g}(b))\not \subseteq M\cap q_2(Z_{3,\mathrm {id}}(b)).$

Since $a>1$ , one may compute that $Y_0\cap Z_{3,\mathrm {id}}(b)=Y\cap Z_{3,\mathrm {id}}(b)$ and it is the set of points $(u_1,u_2,u_3)\in {\mathbb C}^3$ satisfying the following equations:

(8.1) $$ \begin{align} u_1u_2u_3=u_1+u_2+u_3-2 \text{ and } |u_1|=|u_2|=|u_3|=a. \end{align} $$

Consider the function $F: [0,\pi ]^2\to [0,+\infty )$ given by

$$ \begin{align*}F:(\theta_1,\theta_2)\mapsto \left|\frac{a(e^{i\theta_1}+e^{i\theta_2})-2}{a^3e^{i(\theta_1+\theta_2)}-a}\right|.\end{align*} $$

Since $a>1$ , it is well-defined and continuous. We have

$$ \begin{align*}F(0,0)=|(2a-2)/(a^3-a)|=\frac{2}{a(a+1)}<1\end{align*} $$

and

$$ \begin{align*}F(\pi,\pi)=|(-2a-2)/(a^3-a)|=\frac{2}{a(a-1)}>1.\end{align*} $$

There is $\beta \in (0, \pi /2)$ such that for every $\alpha \in [0,\beta ]$ , we have

$$ \begin{align*}F(0,\alpha)<1 \text{ and } F(\pi-\alpha,\pi)>1.\end{align*} $$

Hence, for every $\alpha \in [0,\beta ]$ , there is $\theta (\alpha )\in [0,\pi -\alpha ]$ such that

$$ \begin{align*}F(\theta(\alpha), \theta(\alpha)+\alpha)=1.\end{align*} $$

One may check that

$$ \begin{align*}u_1=ae^{i\theta(\alpha)}, u_2=ae^{i\theta(\alpha)+\alpha}, u_3=a\frac{a(e^{i\theta(\alpha)}+e^{i\theta(\alpha)+\alpha})-2}{a^3e^{i(2\theta(\alpha)+\alpha)}-a}, \alpha\in [0,\beta]\end{align*} $$

are infinitely many distinct solutions of (8.1). So $Y_0\cap Z_{3,\mathrm {id}}(b)$ is infinite.

Since $a>1$ , one may compute that $Y_0\cap Z_{3,g}(b)=Y\cap Z_{3,g}(b)$ , and it is the set of points $(u_1,u_2,u_3)\in {\mathbb C}^3$ satisfying the following equations:

(8.2) $$ \begin{align} u_1u_2u_3=u_1+u_2+u_3-2 \text{ and } u_1\overline{u_2}=|u_3|^2=a^2. \end{align} $$

Consider the function $G: {\mathbb R}_{>0}\times [0,\pi ]\to [0,+\infty )$ given by

$$ \begin{align*}G:(r,\theta)\mapsto \left|\frac{a(r+1/r)e^{i\theta}-2}{a^3e^{2i\theta}-a}\right|.\end{align*} $$

Since $a>1$ , it is well-defined and continuous. We note that $G(1,\theta )=F(\theta ,\theta )$ for $\theta \in [0,\pi ].$ So $G(1,0)<1$ and $G(1,\pi )>1.$ There is $R>1$ such that for every $r\in [1,R]$ , $G(r,0)<1$ and $G(r,\pi )>1.$ Then, for every $r\in [1,R]$ , there is $\theta _r\in [0,\pi ]$ such that $G(r,\theta _r)=1.$

One may check that

$$ \begin{align*}u_1(r)=are^{i\theta_r}, u_2(r)=ar^{-1}e^{i\theta_r}, u_3(r)=a\frac{a(r+1/r)e^{i\theta_r}-2}{a^3e^{2i\theta_r}-a}, r\in [1,R]\end{align*} $$

are infinitely many distinct solutions of (8.1). So $Y_0\cap Z_{3,g}(b)$ is infinite. Moreover, if $r>1$ , then $\max \{|u_1(r)|,|u_2(r)|, |u_3(r)|\}=ar>a$ , so $\{u_1(r), u_2(r),u_3(r)\}\in (M\cap q_2(Z_{3,g}(b)))\setminus (M\cap q_2(Z_{3,\mathrm {id}}(b))).$ This concludes the proof.

Since $M\cap q_2(Z_{3,\mathrm {id}}(b))$ is infinite and $\dim R(M)\cap \Gamma ^{-1}_3(b)=1$ , the Zariski closure of $M\cap q_2(Z_{3,\mathrm {id}}(b))$ in $R(M)$ is $R(M)\cap \Gamma ^{-1}_3(b)$ but $M\cap q_2(Z_{3,\mathrm {id}}(b))\subsetneq (R(M)\cap \Gamma ^{-1}_3(b))({\mathbb R}).$ So $L_1^{-1}(\{a,a,a\})=s_1^{-1}(M\cap q_2(Z_{3,\mathrm {id}}(b)))$ is Zariski dense in $R(s_1)^{-1}(R(M)\cap \Gamma ^{-1}_3(b))$ , where $R(s_1): R(\mathrm {Rat_{2}})\to R(M)$ is induced by $s_1.$ Since $M\cap q_2(Z_{3,\mathrm {id}}(b))\subsetneq (R(M)\cap \Gamma ^{-1}_3(b))({\mathbb R})$ and M is the image of $s_1$ , $L_1^{-1}(\{a,a,a\})\subsetneq R(s_1)^{-1}(R(M)\cap \Gamma ^{-1}_3(b)).$ This concludes the proof.

8.2 Images of algebraic subsets under étale morphisms

Let X be a variety over ${\mathbb R}$ . A closed subset V of $X({\mathbb R})$ is called admissible if there is a morphism $f: Y\to X$ of real algebraic varieties and a Zariski closed subset $V'\subseteq Y$ such that $V=f(V'({\mathbb R}))$ and f is étale at every point in $V'({\mathbb R}).$

Every algebraic subset of $X({\mathbb R})$ is admissible.

Remark 8.13. Denote by J the non-étale locus for f in V. We have $J\cap V({\mathbb R})=\emptyset .$ Since we may replace V by $V\setminus J$ in the above definition, we may further assume that f is étale.

Remark 8.14. Let Y be a Zariski closed subset of X. Since étale morphisms are preserved under base changes, if V is admissible as a subset of $X({\mathbb R})$ , it is admissible as a subset of $Y({\mathbb R}).$

Remark 8.15. An admissible subset is semialgebraic. So it has finitely many connected components.

Proposition 8.16. Let $V_1,V_2$ be two admissible closed subsets of $X({\mathbb R})$ . Then $V_1\cap V_2$ is admissible.

Proof. There is a morphism $f_i: Y_i\to X, i=1,2$ of algebraic varieties and a Zariski closed subset $V_i'\subseteq Y_i$ such that $V_i=f(V_i'({\mathbb R}))$ , and $f_i$ is étale. Then, the fiber product $f: Y_1\times _XY_2\to X$ is étale. Since

$$ \begin{align*}V_1\cap V_2=f_1(V_1'({\mathbb R}))\cap f_2(V_2'({\mathbb R}))=f((V_1'\times_XV_2')({\mathbb R})),\end{align*} $$

$V_1\cap V_2$ is admissible.

The key result in this section is the following, which shows that admissible subsets satisfy the descending chain condition.

Theorem 8.17. Let $V_n, n\geq 0$ be a sequence of decreasing admissible subsets of $X({\mathbb R})$ . Then, there is $N\geq 0$ such that $V_n=V_N$ for all $n\geq N.$

We need the following lemma.

Lemma 8.18. Let V be an admissible closed subset of $X({\mathbb R})$ . Assume that X and $\overline {V}^{\mathrm {zar}}$ are smooth. Then, V is a finite union of connected components of $\overline {V}^{\mathrm { zar}}({\mathbb R}).$

Proof. Since $\overline {V}^{\mathrm {zar}}$ is smooth, different irreducible components of $\overline {V}^{\mathrm {zar}}$ do not meet. So we may assume that $\overline {V}^{\mathrm {zar}}$ is irreducible of dimension $d.$ Hence, $\overline {V}^{\mathrm {zar}}({\mathbb R})$ is smooth; it is of dimension d everywhere.

There is a morphism $f: Y\to X$ of algebraic varieties and a Zariski closed subset $V'\subseteq Y$ such that $V=f(V'({\mathbb R}))$ , and f is étale at every point in $V'({\mathbb R}).$ After replacing $V'$ by $\overline {V'({\mathbb R})}^{\mathrm {zar}}$ , we may assume that $V'({\mathbb R})$ is Zariski dense in $V'.$

For $x\in V$ , there is $y\in V'({\mathbb R})$ such that $V'({\mathbb R})$ has dimension d at y. Since f is étale, $f^{-1}(\overline {V}^{\mathrm {zar}}({\mathbb R}))$ is smooth and of dimension $d.$ Hence, $V'$ coincides with $f^{-1}(\overline {V}^{\mathrm {zar}})$ in some Zariski open neighborhood of y. So $V'({\mathbb R})$ is smooth at y. It follows that f maps some Euclidean neighborhood of y in $V'({\mathbb R})$ to some Euclidean neighborhood of x in $\overline {V}^{\mathrm { zar}}({\mathbb R})$ . This shows that V is open in $\overline {V}^{\mathrm {zar}}({\mathbb R}).$ Then V is a finite union of connected components of $\overline {V}^{\mathrm {zar}}({\mathbb R}).$

Proof of Theorem 8.17

We do the proof by induction on $\dim X$ . When $\dim X=0$ , Theorem 8.17 is trivial.

There is $N\geq 0$ such that $\overline {V_n}^{\mathrm {zar}}$ are the same for $n\geq N.$ After removing $V_n, n=1,\dots , N$ , we may assume that $\overline {V_n}^{\mathrm {zar}}, n\geq 0$ are the same variety. After replacing X by this variety, we may assume that $\overline {V_n}^{\mathrm {zar}}=X$ for all $n\geq 0.$ Let $X_0,X_1$ be the smooth and singular part of X. We only need to show that both $V_n\cap X_0({\mathbb R}), n\geq 0$ and $V_n\cap X_1({\mathbb R}), n\geq 0$ are stable for n large. Since $\dim X_1<\dim X$ , $V_n\cap X_1({\mathbb R}), n\geq 0$ is stable for n large by the induction hypothesis. Since $X_0$ is smooth, by Lemma 8.18, every $V_n$ is a union of connected components of $X_0({\mathbb R}).$ Since $X_0({\mathbb R})$ has at most finitely many connected components, we conclude the proof.

Remark 8.19. Theorem 8.17 does not hold for general semialgebraic subsets. The following example shows that it does not hold even for images of algebraic subsets under finite morphisms. For $n\geq 0$ , set $Z_n:=[n,\infty )\subseteq {\mathbb A}^1({\mathbb R}).$ They are the images of ${\mathbb A}^1({\mathbb R})$ under the finite morphisms $z\mapsto z^2+n, n\geq 0$ . We have $Z_{n+1}\subset Z_n$ but $\cap _{n\geq 0} Z_n=\emptyset $ .

Let $d\geq 2.$ We now view $\text {Rat}_d({\mathbb C})$ as a real variety and study the locus in it with given length spectrum. For $n\geq 1$ , $s=1,\dots , N_n$ and $a\in {\mathbb R}^s/S_s$ , let $\Lambda ^s_n(a)$ be the subset of $t\in \text {Rat}_d({\mathbb C})$ such that $a\subseteq L_n(t)$ (i.e., $f_t^n$ has a subset of fixed points counting with multiplicity, such that the set of norms of multipliers of these fixed points equals to $a).$ It is a closed subset in $\text {Rat}_d({\mathbb C})$ .

Remark 8.20. This notion generalizes the notion $\Lambda _n(a)$ . When $s=N_n$ , we get $\Lambda _n(a)=\Lambda _n^{s}(a).$

Pick $(a_1,\dots , a_s)\in {\mathbb R}^s$ representing $a\in [0,+\infty )^s/ S_s$ . We have

$$ \begin{align*}\Lambda^s_n(a)=\phi^s_n(|\lambda_n^s|^{-1}(a_1,\dots,a_s)).\end{align*} $$

Even though $|\lambda ^s_n|$ is not real algebraic, its square $|\lambda ^s_n|^2$ is real algebraic. So $|\lambda _n^s|^{-1}(a_1,\dots ,a_s)=(|\lambda _n^s|^2)^{-1}(a_1^2,\dots ,a_s^2)$ is real algebraic. Hence, $\Lambda ^s_n(a)$ is semialgebraic. Moreover, if $a_i\neq 1$ for every $i=1,\dots , s$ ,

$$ \begin{align*}|\lambda_n^s|^{-1}(a_1,\dots,a_s)\subseteq (\lambda_n^s)^{-1}(({\mathbb A}^1\setminus \{1\})^s).\end{align*} $$

So $\phi ^s_n$ is étale along $|\lambda _n^s|^{-1}(a_1,\dots ,a_s).$ This shows the following fact.

Proposition 8.21. For $a\in ([0,+\infty )\setminus \{1\})^s/S_s$ , $\Lambda ^s_n(a)$ is admissible.

8.3 Length spectrum

Let f be an endomorphism of ${\mathbb P}^1({\mathbb C})$ of degree $d\geq 2$ . Recall that the length spectrum $L(f)=\{L(f)_n, n\geq 1\}$ of f is a sequence of finite multisets, where $L(f)_n:=L_n(f)$ is the multiset of norms of multipliers of all fixed points of $f^n.$ In particular, $L(f)$ is a multiset of positive real numbers of cardinality $d^n+1$ . For every $n\geq 0$ , let $RL(f)_n$ be the sub-multiset of $L(f)_n$ consisting of all elements $>1.$ We call $RL(f):= \{RL(f)_n, n\geq 1\}$ the repelling length spectrum of f and $RL^*(f):= \{RL^*(f)_n:=RL(f)_{n!}, n\geq 1\}$ the main repelling length spectrum of f. We have $d^{n}+1\geq |RL(f)_n|\geq d^{n}+1-M$ for some $M\geq 0$ . It is clear that the difference $d^{n!}+1-|RL^*(f)_n|$ is increasing and bounded.

Let $\Omega $ be the set of sequences $A_n, n\geq 0$ of multisets consisting of real numbers of norm strictly larger than $1$ satisfying $|A_n|\leq d^{n!}+1$ and for every $a\in A_n$ with multiplicity m, $a^{n+1}\in A_{n+1}$ with multiplicity at least m. For $A,B\in \Omega $ , we write $A\subseteq B$ if $A_n\subseteq B_n$ for every $n\geq 0.$ An element $A=(A_n)\in \Omega $ is called big if $d^{n!}+1-|A_n|$ is bounded. For every endomorphism f of ${\mathbb P}^1({\mathbb C})$ of degree d, we have $RL^*(f)\in \Omega $ and it is big.

For $A\subseteq RL^*(f)$ , by induction, we can show that there is a sequence of sub-multisets $P_n\subseteq \mathrm {Fix}_{n!}(f), n\geq 1$ (here we view $\mathrm {Fix}_{n!}(f)$ as a multiset of cardinal $d^{n!}+1$ ) such that $P_n\subseteq P_{n+1}$ and $A_n=\{|df^{n!}(x)||\,\,x\in P_n\}$ . Such $P:=(P_n)$ is called a realization of A, which may not be unique. Further, assume that A is big. Then, for every realization of A, $|\mathrm {Fix}_{n!}(f)\setminus P_n|$ is bounded. It follows that $\mathrm { Per}\,(f)\setminus (\cup _{n\geq 0}P_n)$ is finite.

Let $A\in \Omega .$ Define $\Lambda (A):=\cap _{n\geq 1}\Lambda ^{|A_n|}_{n!}(A_n),$ which is the locus of $t\in \text {Rat}_d$ satisfying $A\subseteq RL^*(f_t)$ . It is clear that $\Lambda ^{|A_n|}_{n!}(A_n), n\geq 1$ is decreasing, and by Proposition 8.21, each of them is admissible. Hence, by Theorem 8.17, we get the following result.

Proposition 8.22. There is $N(A)\geq 0$ such that

$$ \begin{align*} \Lambda(A)=\Lambda^{|A_{N(A)}|}_{N(A)!}(A_{N(A)}), \end{align*} $$

which is admissible.

Let $\gamma \simeq [0,1]$ be a real analytic curve in $\text {Rat}_d({\mathbb C})$ . We view $\gamma \times {\mathbb P}^1({\mathbb C})$ as a subset of $\text {Rat}_d({\mathbb C})\times {\mathbb P}^1({\mathbb C})$ . Let $f_{\gamma }$ be the restriction of $f_{\text {Rat}_d({\mathbb C})}$ to $\gamma \times {\mathbb P}^1({\mathbb C})$ . For every n-periodic point $x=(t,y)\in \gamma \times {\mathbb P}^1({\mathbb C})$ , let $\gamma ^n_x$ be the connected component of

$$ \begin{align*} (\gamma\times {\mathbb P}^1({\mathbb C}))\cap \text{Rat}_d({\mathbb C})[n]=\phi_n^{-1}(\gamma) \end{align*} $$

containing $x.$

Remark 8.23. If x is repelling for $f_t$ , then $\phi _n$ is étale at $(t,x)$ . Hence, it induces an isomorphism from some neighborhood of $(x,t)$ in $\gamma ^n_x$ to its image in $\gamma .$

Moreover, if $|\lambda _n|(\gamma ^n_x)\subseteq (1,+\infty )$ , then $\phi _n$ is étale along $\gamma ^n_x$ . In particular, $\phi _n|_{\gamma ^n_x}: \gamma ^n_x\to \gamma $ is a covering map. Since $\gamma $ is simply connected, $\phi _n|_{\gamma ^n_x}: \gamma ^n_x\to \gamma $ is an isomorphism. If $n| m$ , then $\gamma ^n_x\subseteq \gamma ^m_x$ . However, for every $(u,y)\in \gamma ^n_x$ , the multiplicity of y in $\mathrm {Fix}(f^m_u)$ is $1.$ So $\gamma ^m_x$ coincide with $\gamma ^n_x$ in a neighborhood of $y.$ Hence, $\gamma ^m_x=\gamma ^n_x$ . This implies that every $y\in \gamma _x$ has the same minimal period and for every period l of y, $\gamma _y^l=\gamma _x^n.$

Lemma 8.24. Fix $A\in \Omega $ . Assume that for every $t\in \gamma $ , $A\subseteq RL^*(f_t)$ . Then, there is a realization P of A for $f_0$ , such that the following holds:

1. (i) For every $x\in \cup _{n\geq 0}P_n$ , $\gamma ^{m}_{(0,x)}$ does not depend on the choice of period m of $x.$ We denote by $\gamma _x=\gamma ^{m}_{(0,x)}$ for some (then every) period m of x. Then $\phi _m|_{\gamma _x}: \gamma _x\to \gamma $ is a homeomorphism and it is étale along $\gamma _x.$ In particular, for different points x, $\gamma _x$ are disjoint.

2. (ii) For every $x\in \cup _{n\geq 0}P_n$ , with a period m, $|\lambda _m|$ is a constant on $\gamma _x.$

Proof. For every $n\geq 1$ , let $B_n$ be the subset of $\mathrm {Fix}(f_0^{n})$ such that $|\lambda _n|$ is a constant $>1$ on $\gamma _{(0,x)}^n$ . If $x\in B_n$ for some $n\geq 1$ , by Remark 8.23, $x\in B_m$ for every period m of x and $\gamma _x:=\gamma ^m_{(0,x)}$ does not depend on the choice of period $m.$ Moreover, $\phi _m|_{\gamma _x}: \gamma _x\to \gamma $ is a homeomorphism and it is étale along $\gamma _x.$ In particular, for different points x, $\gamma _x$ are disjoint.

It is clear that $B=(B_{n!})$ realizes an element $C\in \Omega .$ We only need to show that $A\subseteq C.$ Let a be an element in $A_n$ of multiplicity $l\geq 1.$ Then, for every $t\in \gamma $ , since $|a|>1$ , $|\lambda _{n!}|^{-1}(a)\cap \phi _{n!}^{-1}(t)$ contains at least l distinct points. Let $x_1,\dots , x_s$ be the elements in $x\in B_{n!}$ with $\lambda _{n!}((0,x))=a.$ We only need to show that $s\geq l.$ For every $i=1,\dots ,s$ , $\gamma _{x_i}$ is a connected component of $\phi _{n!}^{-1}(\gamma ).$ Set $Z:=\phi _{n!}^{-1}(\gamma )\setminus \cup _{i=1}^s\gamma _{x_i}.$ If $s<l$ , then for every $t\in \gamma $ , $Z\cap |\lambda _{n!}|^{-1}(a)\cap \phi _{n!}^{-1}(t)$ has at least one point. So there is $y\in Z$ such that $\gamma ^{n!}_z\cap |\lambda _{n!}|^{-1}(a)$ is infinite. Since both $\gamma ^{n!}_z$ and $|\lambda _{n!}|^{-1}(a)$ are real analytic in $\gamma \times {\mathbb P}^1({\mathbb C})$ , $\gamma ^{n!}_z\subseteq |\lambda _{n!}|^{-1}(a).$ By Remark 8.23, $\gamma ^{n!}_z$ meets $\phi _{n!}^{-1}(0)$ at some point $(0,x)$ for some $x\in B_n.$ So $\gamma ^{n!}_z=\gamma _x$ , which is a contradiction.

8.4 Length spectrum as moduli

Let $\Psi : \text {Rat}_d({\mathbb C})\to {\mathcal M}_d({\mathbb C})=\text {Rat}_d({\mathbb C})/\mathrm {PGL}\,_2({\mathbb C})$ be the quotient map. Let $FL_d({\mathbb C})\subseteq \text {Rat}_d({\mathbb C})$ be the locus of Lattès maps, which is Zariski closed in $\text {Rat}_d({\mathbb C})$ . We now prove Theorem 1.5 via proving the following stronger statement.

Theorem 8.25. If $A\in \Omega $ is big, then $\Phi (\Lambda (A)\setminus FL_d({\mathbb C}))\subseteq {\mathcal M}_d$ is finite.

Proof. By Proposition 8.22, $\Lambda (A)$ is admissible in $\text {Rat}_d({\mathbb C})$ . Hence $\Lambda (A)\setminus FL_d({\mathbb C})$ is admissible in $\text {Rat}_d({\mathbb C})\setminus FL_d({\mathbb C})$ . In particular, $\Lambda (A)\setminus FL_d({\mathbb C})$ and $\Phi (\Lambda (A)\setminus FL_d({\mathbb C}))$ are semialgebraic.

To get a contradiction, assume that $\Phi (\Lambda (A)\setminus FL_d({\mathbb C}))$ is not finite. By Nash Curve Selection Lemma [BCR98, Proposition 8.1.13], there is a real analytic curve $\gamma \simeq [0,1]$ in $\Lambda (A)\setminus FL_d({\mathbb C})$ whose image in ${\mathcal M}_d$ is not a point. Since non-flexible Lattès exceptional endomorphisms are isolated in the moduli space $\mathcal {M}_d$ , there is at least one $f_t$ that is not exceptional. Without loss of generality we assume $f_0$ is not exceptional. We now apply Lemma 8.24 for $\gamma $ and A, and follows the notation there. Set $Q:=\cup _{n\geq 0}P_n$ . Then $S:=\mathrm {Per}\,(f_0)\setminus Q$ is finite.

Pick any $z_0\in Q.$ By the discussion in Example 7.3, there exists a horseshoe K of $f_0$ containing $z_0$ and $K\cap S=\emptyset $ . There is $m\geq 0$ such that $f_0^m(K)=K$ and $f_0^m(z_0)=z_0$ . By Lemma 6.1, there exists $\varepsilon>0$ and a continuous map $h:[0,\varepsilon ]\times K\to {\mathbb P}^1({\mathbb C})$ such that for each $t\in [0,\varepsilon ]$ :

1. (i) $K_t:=h(t,K)$ is an expanding set of $f_t^m$ .

2. (ii) the map $h_t:=h(t, \cdot ):K\to K_t$ is a homeomorphism and $f_t^m\circ h_t=h_t\circ f_0^m$ .

For every $t\in [0,\varepsilon ]$ and for every $w_0\in K$ satisfying $f_0^{nm}(w_0)=w_0$ , we have $f_t^{nm}(h_t(w_0))=h_t(w_0)$ . It follows that $h_t(w_0)=\gamma _{w_0}(t).$ Since $|\lambda _{nm}|$ is a constant on $\gamma _{w_0}$ , we get $|df_0^{nm}(w_0)|=|df_t^{mn}(h_t(w_0))|$ . We claim that $K_t$ is a CER of $f_t$ . We check that $(f_t,K_t)$ satisfies Definition 7.1: since $K_t$ is expanding by Lemma 6.1, (ii) holds; since topological exactness and openness are preserved by topological conjugacy, by Remark 7.2), (i) and (iii) hold.

Since $f_0$ is not exceptional, by Theorem 1.1, K is a nonlinear CER for $f_0$ . By Theorem 7.6, for every fixed $t\in [0,\varepsilon ]$ , the conjugacy $h_t$ can be extended to a conformal map $h_t:U\to V$ where U is a neighborhood of K and V is a neighborhood of $K_t$ . This implies $df_0^m(z_0)=df_t^m(\gamma _{z_0}(t))(=df_t^m(h_t(z_0)))$ or $df_0^m(z_0)=\overline {df_t^m(\gamma _{z_0}(t))}$ . Since $df_t^m(\gamma _{z_0}(t))$ depends continuously on t, we must have $df_0^m(z_0)=df_t^m(\gamma _{z_0}(t))$ when $t\in [0,\varepsilon ]$ . Since $\gamma _{z_0}$ is real analytic, the map $t\mapsto df_t^m(\gamma _{z_0}(t))$ is real analytic on $\gamma =[0,1]$ . It is a constant on $[0,\varepsilon ]$ . Hence, it is a constant on $\gamma .$ Let n be any period of $z_0$ . The above argument shows that $(\lambda _n|_{\gamma _{z_0}})^m$ is a constant. Hence, $\lambda _n|_{\gamma _{z_0}}$ is a constant.

Since our choice of $z_0\in Q$ is arbitrary, for every $z_0\in Q$ , of period n, the map $t\mapsto df_t^n(\phi (t))$ is a constant on $[0,1]$ . Since S is finite, all $f_t$ have the same multiplier spectrum for periodic points of sufficiently high period.

The set of all endomorphisms in $\text {Rat}_d({\mathbb C})$ with the same multiplier spectrum of $f_0$ for periodic points with period at least $N\geq 1$ is an algebraic variety. We denote it by $V_N$ . There exists $N\geq 1$ such that $\gamma \subseteq V_N.$ Furthermore, there exists an irreducible component X of $V_N$ which contains $\gamma .$ The irreducible variety X forms a stable family (see [McM16, Chapter 4]), since the period of attracting cycles are bounded in $V_N$ . The variety X is not isotrivial since $\Psi (\gamma )$ is not a point. By Theorem 1.2, $\gamma \subseteq X$ is contained in the flexible Lattès family, which is a contradiction.