Proper extensions of the 2-sphere’s conformal group present entropy and are 4-transitive

Abstract

In this paper, we prove using elementary techniques that any group of diffeomorphisms acting on the 2-sphere and properly extending the conformal group of Möbius transformations must be at least 4-transitive or, more precisely, arc 4-transitive. As an important consequence, we derive that any such group must always contain an element of positive topological entropy. We also provide a self-contained characterization, in terms of transitivity, of the Möbius transformations within the full group of sphere diffeomorphisms.

1 Introduction

Let M be a closed and oriented topological manifold, and consider the set $\mathrm {Homeo}_{}(M)$ of all its orientation-preserving homeomorphisms. The usual uniform metric turns this set—endowed with the composition operation—into a topological group, the subgroups of which one may try to understand and classify.

In his early 2000s’ essay [5], Ghys proposed such a classification for closed and transitive groups acting on the unit circle. Here, closed refers to the uniform topology mentioned above, while transitive means that any given point p can be mapped onto another given point q via some transformation in the group. The corresponding result, which we quote later on this paper, was proven in 2006 by Giblin and Markovic, see [6].

A relevant part of understanding the closed subgroups of $\mathrm {Homeo}_{}(M)$ is to deal not only with their inclusions, but also with questions of maximality. In other words, determining whether or not between a given subgroup and the full group of homeomorphisms one may find proper intermediate subgroups, up to their closures. For example, Le Roux proved in [10] that, in triangulable manifolds of dimension two or higher, the group of area-preserving homeomorphisms is always maximal.

Specializing in the 2-sphere 𝕊², F. Kwakkel and the second author derived in [9] a number of results concerning subgroups of $\mathrm {Homeo}_{}(\mathbb {S}^2)$ related to key symmetries, one of which is ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ : the Möbius group of conformal diffeomorphisms. Among others, they left open the question of whether there is no proper intermediate group of diffeomorphisms between ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ and $\mathrm {Homeo}_{}(\mathbb {S}^2)$ , up to uniform closure. (Due to a difference of opinion between its authors, [9] remains as an unsubmitted preprint, of which we refer to version 2 in ArXiv. This note is self-contained, and makes no direct use of the results therein.)

This question is known to have a positive answer in the case of the circle, where the proof is related to higher orders of transitivity. In this paper, we provide the following insight into extensions of ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ , from the transitivity viewpoint.

Theorem A. Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, its identity component is arc 4-transitive. In particular, G is at least 4-transitive.

Above, the stronger concept of arc-transitivity is introduced. It means that not only ordered lists $( p_1, \ldots , p_4 )$ and $( q_1, \ldots , q_4 )$ can be mapped one onto another by a transformation in G, each q _i is actually the endpoint of p _i ’s trajectory under a isotopy in G starting at the identity. Theorem A—or rather its proof—has an interesting dynamical implication: the constructions used to derive it also allow one to deduce the presence of an element having positive topological entropy in G.

Theorem B. Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, G contains an element f fixing at least four points, and such that its restriction to the complement of these points is isotopic to a pseudo-Anosov map relative to them. In particular, f has strictly positive topological entropy.

Theorem B could, in principle, be derived from Theorem A along with an abstract result of [9]. However, we favor the more explicit construction presented here, as both rely on Nielsen–Thurston classification theory. We also include an interesting result—albeit not new—stemming from the techniques used to show Theorems A and B. It is stated in terms of the key definition below.

Definition 1.1. Given $k \in \mathbb {N}$ , the action of a subgroup $G \subset \mathrm {Homeo}_{}(M)$ is said to be k-transitive if, for every pair of k-tuples $( p_1, \ldots , p_k )$ and $( q_1, \ldots , q_k )$ , each of them consisting of mutually distinct points, there exists some transformation $g \in G$ such that $q_i = g(p_i)$ for each $i \in \lbrace 1, \ldots , k \rbrace $ . If, in addition, such g is unique, the action is said to be sharply k-transitive.

It is well known that ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ is sharply 3-transitive. As it turns out, this is a defining property: in [9], Kwakkel and the second author show that a sharply 3-transitive subgroup of ${\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ extending the group of rotations must be ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Since the cited work has not been published, we present for completeness a new simple, independent and—in our opinion—rather amusing proof of this fact, as stated below.

Theorem C. Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group extending the rotations group ${\mathrm {Rot}}(\mathbb {S}^2)$ . Then, G sharply 3-transitive implies $G={\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ .

1.1 Preliminaries

Let us start by terminology. As aforementioned, $\mathrm {Homeo}_{}(M)$ denotes the set of orientation-preserving homeomorphisms of a closed and oriented topological manifold M, on which the uniform metric is well defined. If M is smooth, we may also consider the subset ${\mathrm {Diff}}^{1}_{}(M) \subset \mathrm {Homeo}_{}(M)$ of orientation-preserving diffeomorphisms of class C ¹ and its finer C ¹ topology, which takes into account the local expressions of differentials as well.

Consider the circle $M = \mathbb {S}^1$ , which can be thought of as the set of complex numbers of unit absolute value or as the compactification $\mathbb {R}\cup \lbrace \infty \rbrace $ . Each such description is linked to a canonical group acting on it. In the first case, we have the rotations group ${\mathrm {Rot}}(\mathbb {S}^1)$ , which is naturally identified with the circle itself, thus being a compact Lie group. In the second case, we have the Möbius group ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^1)$ , obtained by the action of ${\mathrm {PSL}}( 2; \mathbb {R} )$ on the extended real line via linear fractional transformations.

The rotations group can be realized as a proper subgroup of the Möbius group. Interestingly enough, these two yield a full description of the closed and transitive subgroups of $\mathrm {Homeo}_{}(\mathbb {S}^1)$ , which is the content of the following theorem, by Giblin and Markovic.

Theorem. (Giblin and Markovic [6]).

Let G be a closed and transitive subgroup of $\mathrm {Homeo}_{}(\mathbb {S}^1)$ containing a non-trivial path connected component. Then, G is conjugate to one, and only one, of the following:

1. (i) ${\mathrm {Rot}}(\mathbb {S}^1)$ ;

2. (ii) ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{k} (\mathbb {S}^1)$ ; or

3. (iii) $\mathrm {Homeo}_{k} (\mathbb {S}^1)$ ,

where the subscript $k \in \mathbb {N}$ indicates cyclic cover of order k. Furthermore, ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^1)$ is maximal: there is no closed subgroup properly containing ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^1)$ and properly contained in $\mathrm {Homeo}_{}(\mathbb {S}^1)$ .

Consider now the 2-sphere $M = \mathbb {S}^2$ . It can be thought of either as the points of Euclidean 3-space at unit distance from the origin or as the compactification $\mathbb {C}\cup \lbrace \mathbf {\infty } \rbrace $ . In the latter case, identification is provided by stereographic projection from the North Pole $( 0,0,1 )$ . We denote points on the plane and their stereographic images by the same letters and confound them without notice. Any sphere map fixing ∞ thus defines, by stereographic conjugation, a planar map with the same degree of regularity, also denoted by the same letter.

The group of sphere rotations ${\mathrm {Rot}}(\mathbb {S}^2)$ consists of Euclidean isometries preserving both 𝕊² and orientation. It is realized by ${\mathrm {SO}}( 3; \mathbb {R} )$ , and is thus a compact Lie group. Each such transformation amounts to prescribing an axis and a rotation angle around that axis. From these facts, ${\mathrm {Rot}}(\mathbb {S}^2)$ is seen to be closed and transitive. Rotations are minimal in the following sense: a compact group $G \subset \mathrm {Homeo}_{}(\mathbb {S}^2)$ must be topologically conjugate to a closed subgroup of ${\mathrm {Rot}}(\mathbb {S}^2)$ —a result from the 1940s by Kerékjártó. A proof according to contemporary standards is given by Kolev in [8]. If G is further transitive, then it is actually conjugate to ${\mathrm {Rot}}(\mathbb {S}^2)$ . The problem of classifying the groups contained between ${\mathrm {Rot}}(\mathbb {S}^2)$ and $\mathrm {Homeo}_{}(\mathbb {S}^2)$ is thus called the kernel subgroup problem, and such intermediate groups containing the rotations are hereafter called homogeneous, following the terminology established in [9].

The Möbius group ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ is an example of a homogeneous group. It is defined similarly to its circle counterpart, by the action of ${\mathrm {PSL}}( 2; \mathbb {C} )$ on the extended plane via linear fractional transformations. More precisely, one associates to (the class of) the matrix A a mapping M _A as follows:

$$ \begin{align*}\text{if } A = \pm \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\mathrm{PSL}}( 2; \mathbb{C} )\ \text{then } M_A (z) = \frac{a \, z + b}{c \, z + d} \quad\text{for every } z \in \mathbb{C}\cup \lbrace \infty \rbrace \text{.}\end{align*} $$

This procedure characterizes the conformal diffeomorphisms of 𝕊² endowed with its canonical smooth structure. The Möbius group is thoroughly understood—see [4] and [12, Ch. 3]. In particular, spherical rotations are induced by the subgroup ${\mathrm {PSU}}( 2; \mathbb {C} )$ . Le Roux, Kwakkel, and the second author left open the question of whether the inclusion of ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ in $\mathrm {Homeo}_{}(\mathbb {S}^2)$ is maximal. Our contribution to this problem is Theorem A above.

It should be stressed that, in the circle setting, 4-transitivity is sufficient to ensure k-transitivity for any $k> 4$ . However, the argument used in [6] to do so presents no obvious generalization to higher dimensions, for it relies on the complement of a finite subset of 𝕊¹ being composed of disjoint open intervals.

The constructions in this paper are mostly based upon isotopies, here understood as families $( f_t )_{t \in I}$ for which $t \mapsto f_t$ is continuous with respect to the uniform metric. When the interval in question is the standard unit interval, we denote it by $\mathbb {I}= [ 0,1 ]$ . Often, it will be the case that $f_t \in {\mathrm {Diff}}^{1}_{}(M)$ for every $t \in I$ , but $t \mapsto f_t$ can only be assured to be continuous with respect to the uniform metric. To avoid confusion, isotopies for which this association is actually continuous with respect to the C ¹ topology will be explicitly referred to as diffeotopies.

Definition 1.2. Let $G \subset \mathrm {Homeo}_{}(M)$ be a subgroup. An isotopy $( f_t )_{t \in I}$ such that $0 \in I$ , $f_0 = \mathrm {id}$ , and $f_t \in G$ for every $t \in I$ will be referred to as an $\mathcal {I}G$ -isotopy. Given a point $x \in M$ , we define its trajectory under f as

$$ \begin{align*}\gamma _{f} ( x ) = \lbrace f_t (x) : t \in I \rbrace \text{.}\end{align*} $$

It will often be thought of as a path, oriented according to its natural direction of travel along increasing values of t. If I is unbounded above, we further define the ω-limit as the following set of accumulation points:

$$ \begin{align*}\omega _{f} ( x ) = \lbrace y \in M : \text{ there exists a sequence } t_n \nearrow + \infty \text{ such that } f_{t_n}(x) \rightarrow y \rbrace \text{.}\end{align*} $$

There is an analogous notion of α-limit when I is unbounded below. Lastly, G is said to be arc k-transitive if for every pair of k-tuples, as in Definition 1.1, there exists an $\mathcal {I}G$ -isotopy $( g_t )_{t \in \mathbb {I}}$ such that $g_1(p_i) = q_i$ for each $i \in \lbrace 1, \ldots , k \rbrace $ .

Despite analogies between isotopy trajectories and flows being very limited, such suggestive terminologies prove themselves pictorially useful in arguments to come. The reason is the concept we now develop, which plays vaguely the same role as that of the semigroup property for flows.

Definition 1.3. For a fixed subgroup $G \subset \mathrm {Homeo}_{}(M)$ and given $z,w \in M$ , say that

$$ \begin{align*}z \sim _{G}w \iff \text{ there exists an } \mathcal{I}G\text{-isotopy } ( f_t )_{t \in \mathbb{I}} \text{ in } G \text{ such that } f_1 (z) = w\text{.}\end{align*} $$

This is an equivalence relation, under which the class of a point $z \in M$ is denoted by $\mathcal {A}_{G}( z )$ , and referred to as the points accessible from z (in G).

Lemma 1.4. Let $( f_t )_{t \in I}$ and $( h_t )_{t \in J}$ be $\mathcal {I}G$ -isotopies, where I and J are intervals of any kind. Then, for any two $z,w \in M$ , $\gamma _{f} ( z ) \cap \gamma _{h} ( w ) \ne \emptyset $ implies $\mathcal {A}_{G}( w ) = \mathcal {A}_{G}( z )$ .

Proof. Assume $f_a (z) = h_b (w)$ for some $a \in I$ and $b \in J$ . Given $y \in \mathcal {A}_{G}( w )$ , there exists an $\mathcal {I}G$ -isotopy $( g_t )_{t \in \mathbb {I}}$ such that $g_1 (w) = y$ . Thus,

$$ \begin{align*}k_t :={\begin{cases} f_{3a t} & \text{if } 0 \leq t \leq {1}/{3} \text{,} \\ h_{( 2 - 3t ) b} \circ h_b^{-1} \circ f_a & \text{if } {1}/{3} \leq t \leq {2}/{3} \text{,} \\ g_{3t-2} \circ h_b^{-1} \circ f_a & \text{if } {2}/{3} \leq t \leq 1 \text{,} \end{cases}}\end{align*} $$

is an $\mathcal {I}G$ -isotopy satisfying $k_1 (z) = y$ . This shows that $\mathcal {A}_{G}( w ) \subset \mathcal {A}_{G}( z )$ , and the converse inclusion follows by symmetry.

We finish this introduction fixing some references: ∞ is the North Pole, 0 is the South Pole, corresponding to the plane’s origin, and 1 is the point $(1,0,0)$ , corresponding to its counterpart on the real axis. The meridian through 0, 1 and ∞ is denoted by Γ. It defines on its left a western hemisphere $\mathcal {H}^{+}$ , corresponding to the upper half-plane, and on its right an eastern hemisphere $\mathcal {H}^{-}$ . For any group G, we also adopt the following notation for stabilizers:

$$ \begin{align*}G_1 :={\mathrm{Stab}}_{G} \lbrace \mathbf{\infty } \rbrace, \quad G_2 : = {\mathrm{Stab}}_{G} \lbrace \mathbf{0}, \mathbf{\infty } \rbrace \quad \text{and} \quad G_3 : = {\mathrm{Stab}}_{G} \lbrace \mathbf{0}, \mathbf{1}, \mathbf{\infty } \rbrace \text{.}\end{align*} $$

1.2 Paper outline

Given a group G properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ , we consider the subgroups G _k , $1 \leq k \leq 3$ , as above. Our objective is to conclude that G ₃ is (one) transitive. We begin by establishing an Extension Lemma at the end of §2, which states that G ₂ must contain an isotopy between the identity and a map having a hyperbolic saddle point at the (fixed) South Pole.

From this starting point, we consider the reference meridian Γ and promote two parallel processes. In §3, we show that every point not in Γ admits a full time isotopy in G ₃ under which its trajectory accumulates at the poles. In §4, we obtain a finite time isotopy in G ₃ under which a point is moved from one side of the meridian to the other.

In §5, we show our main theorems. Theorem A is derived upon combining isotopies of the types just described and concluding that all but three points of the sphere are arc connected in the sense of Definition 1.2. Theorem B is also derived by convenient combinations of segments of such isotopies, but to produce a ‘topological figure-8’, a device that implies positive entropy due to the Nielsen–Thurston classification theory.

In §6, we independently derive Theorem C. First, a purely topological argument shows that, if G is a sharply 3-transitive and homogeneous group of homeomorphisms, then G ₂ must permute parallels. This fact, when combined with differentiability, yields conformality—first at the poles, and then at every point.

2 Extensions of Möbius

For a given $0 \leq \theta < 2 \pi $ , we let $R_{\theta }$ denote the counterclockwise rotation of angle θ around the axis from 0 to ∞. If it is thought of as a planar map, it may be written as $R_{\theta } (z) = \mathrm {e}^{\mathrm {i} \,\theta } z$ . Also, for a given $\rho> 0$ , we let $H_{\rho } \in {\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{2} (\mathbb {S}^2)$ denote the transformation induced by the planar homothety $z \mapsto \rho z$ . Since the tangent space $T_{\mathbf {0}} \mathbb {S}^2$ can be identified with the horizontal subspace $\mathbb {R}^2 \times \lbrace 0 \rbrace \simeq \mathbb {R}^2$ , we may write $\mathrm {D} R_{\theta } ( \mathbf {0} ) = R_{\theta }$ and $\mathrm {D} H_{\rho } ( \mathbf {0} ) = \rho \, \mathrm {id}$ , in a slight abuse of notation.

Lemma 2.1. Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, there exists $\hat {g} \in G_2$ for which $\mathrm {D} \hat {g} ( \mathbf {0} ) = {\mathrm {diag}} [ \unicode{x3bb} , \unicode{x3bb} ^{-1} ]$ with respect to the canonical basis, where $0 < \unicode{x3bb} < 1$ .

Proof. Since G is a proper extension of the conformal group, it contains at least one map $\hat {h}$ which is non-conformal at some point of the sphere. By precomposing and postcomposing with suitable Möbius transformations, it can be assumed that such map lies in G ₂ (that is, it fixes the poles) and that $A = \mathrm {D} \hat {h} ( \mathbf {0} )$ is non-conformal.

This means that there exists a pair of unit vectors $u,v$ such that ${\mathrm {ang}}( Au, Av ) \ne {\mathrm {ang}}( u,v )$ . Let $R_{(1)}$ be a planar rotation such that $R_{(1)} ( Au )$ is a positive multiple of u. Then, $A_1 = R_{(1)} A$ has an eigenvalue $\unicode{x3bb} _1> 0$ for which u is a unit eigenvector. Also, λ₁ cannot be of maximal geometric multiplicity, leaving two possibilities.

1. • If A ₁ is defective, let $\lbrace u,w \rbrace $ form an orthogonal chain of generalized eigenvectors. That is, $A_1 w = u + \unicode{x3bb} _1 w$ and $\langle u , w \rangle = 0$ , where $\langle \cdot , \cdot \rangle $ is the Euclidean inner product. For each $\phi \in [ 0,\pi /2 ]$ , consider

   $$ \begin{align*}x_{\phi } :=\cos \phi \; u + \sin \phi \; \frac{w}{|w |} \quad \text{and} \quad y_{\phi } :=- \sin \phi \; u + \cos \phi \; \frac{w}{|w |} \text{.}\end{align*} $$

   Then, $A_1 x_{{\phi }_0}$ and $A_1 y_{{\phi }_0}$ are orthogonal for some $0 < {\phi }_0 < \pi / 2$ , since

   $$ \begin{align*}0 < \langle A_1 x_0 , A_1 y_0 \rangle = - \langle A_1 x_{\pi / 2} , A_1 y_{\pi / 2} \rangle \text{.}\end{align*} $$

   Define $\hat {x} :=x_{{\phi }_0}$ and $\hat {y} :=y_{{\phi }_0}$ . Since A ₁ preserves orientation, the orthonormal frame $\lbrace A_1 \hat {x} / | A_1 \hat {x} | , A_1 \hat {y} / | A_1 \hat {y} | \rbrace $ can be applied onto $\lbrace \hat {x}, \hat {y} \rbrace $ by a planar rotation $R_{(2)}$ . Therefore, $R_{(2)} ( A_1 \hat {x} ) = \nu _1 \, \hat {x}$ and $R_{(2)} ( A_1 \hat {y} ) = \nu _2 \, \hat {y}$ , where (say) $0 < \nu _1 < \nu _2$ .

   Lastly, let R be a planar rotation such that $R^{-1} \, \hat {x} = \partial / \partial x$ , $\rho :=(\nu _1 \nu _2)^{-1/2}$ , $\unicode{x3bb} :=(\nu _1 / \nu _2)^{1/2} < 1$ , and $\hat {A} :=\rho \, R^{-1} R_{(2)} A_1 R$ . Then, $\hat {A}$ has the form described in the lemma’s statement with respect to the canonical basis. Also, $\hat {A} = \mathrm {D} \hat {g} ( \mathbf {0} )$ , where $\hat {g} = R^{-1} \circ H_{\rho } \circ R_{(2)} \circ R_{(1)} \circ \hat {h} \circ R \in G_2$ . This completes the proof in this case.

2. • If A ₁ has a second eigenvalue $\unicode{x3bb} _2 \ne \unicode{x3bb} _1$ , fix a positive basis of unit eigenvectors $\lbrace u,w \rbrace $ and let ${w^{\perp }}$ be the orthogonal complement of w with respect to u. Letting

   $$ \begin{align*}x_{\phi } :=\cos \phi \; u + \sin \phi \; \frac{{w^{\perp }}}{|{w^{\perp }}|} \quad \text{and } \quad y_{\phi } :=- \sin \phi \; u + \cos \phi \; \frac{{w^{\perp }}}{|{w^{\perp }}|} \text{,}\end{align*} $$
   the proof follows as before, but now $R_{(2)}$ may be the identity if $w = {w^{\perp }}$ .

Lemma 2.2. (Extension Lemma)

Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, there exists an $\mathcal {I}G_2$ -diffeotopy $( g_t )_{t \in \mathbb {I}}$ such that:

1. (1) for every $t> 0$ , the differential $\mathrm {D} g_t ( \mathbf {0} )$ is a hyperbolic saddle, having the tangent line $T_{\mathbf {0}} \Gamma $ as its stable direction;

2. (2) $\mathrm {D} g_1 ( \mathbf {0} ) = {\mathrm {diag}} [ \unicode{x3bb} , \unicode{x3bb} ^{-1} ]$ with respect to the canonical basis of $T_{\mathbf {0}} \mathbb {S}^2$ , where $0{\kern-1.5pt} <{\kern-1.5pt} \unicode{x3bb}{\kern-1.5pt} <{\kern-1.5pt} 1$ .

Proof. Lemma 2.1 yields $\hat {g} \in G_2$ such that $\hat {A} :=\mathrm {D} \hat {g} ( \mathbf {0} ) = {\mathrm {diag}} [ \mu , \mu ^{-1} ]$ , $0 < \mu < 1$ . For $s \in [ 0, \pi /2 ]$ , we let $B_s :=\hat {A}^{-1} \, R_s \, \hat {A}$ , $v_s :=B_s ( \partial / \partial x )$ , and consider the continuous function $\theta (s)$ given by the angle between v _s and $\partial / \partial x$ . If $C_s :=R_{- \theta (s)} \, B_s$ , then $C_0 = \mathrm {id}$ and C _s has the x-axis as a contracting direction of rate $\unicode{x3bb} _s = |v_s | < 1$ for $s>0$ . Since C _s has unit determinant, it is actually a hyperbolic saddle. In general it may not be diagonal, but we explicitly compute $C_{\pi /2} = {\mathrm {diag}} [ \unicode{x3bb} , \unicode{x3bb} ^{-1} ]$ , where $\unicode{x3bb} = \mu ^2$ . Lastly, we define $( g_t )_{t \in \mathbb {I}}$ as

(1) $$ \begin{align} g_t = R_{- \theta ( {\pi t}/{2} ) } \circ \hat{g}^{-1} \circ R_{{\pi t}/{2}} \circ \hat{g} \in G_2 \text{.}\end{align} $$

Since θ is continuous, equation (1) is continuous with respect to the C ¹ topology. For each t, we have $\mathrm {D} g_t ( \mathbf {0} ) = C_{{\pi t}/{2}}$ . Given that the x-axis corresponds to Γ—with the direction generated by $\partial / \partial x$ identified with $T_{\mathbf {0}} \Gamma $ —the considerations previously made translate into the statements of the lemma, completing the proof.

3 A fundamental lemma

For each angle $\alpha \in ( 0, \pi /2 )$ , consider the closed cone

$$ \begin{align*}\mathcal{C}_{\alpha } = \lbrace r \mathrm{e}^{ \mathrm{i} \,\theta } : r \geq 0 \text{ and either } |\theta | \leq \alpha \text{ or } |\theta - \pi | \leq \alpha \rbrace \text{.}\end{align*} $$

If $0 < \unicode{x3bb} < 1$ , any such cone is ‘broadened’ under the action of the hyperbolic matrix $A = {\mathrm {diag}} [ \unicode{x3bb} , \unicode{x3bb} ^{-1} ]$ , $0 < \unicode{x3bb} < 1$ . More precisely, $A ( \mathcal {C}_{\alpha } ) = \mathcal {C}_{\alpha + 2\tau }$ , for some $\tau> 0$ . Approximating the polar angle of $g(z)$ by that of $\mathrm {D} g ( \mathbf {0} )z$ then implies the following lemma.

Lemma 3.1. Let g be a planar diffeomorphism for which $\mathrm {D} g ( \mathbf {0} ) = {\mathrm {diag}} [ \unicode{x3bb} , \unicode{x3bb} ^{-1} ]$ , $0{\kern-1.5pt} <{\kern-1.5pt} \unicode{x3bb}{\kern-1.5pt} <{\kern-1.5pt} 1$ . Then, for a given $0 < \alpha < \pi / 2$ , there exist $\tau> 0$ and $\delta> 0$ satisfying:

$$ \begin{align*}0 < |z | < \delta \quad \text{and} \quad z \notin \mathcal{C}_{\alpha } \text{ imply } R_{\omega } ( g(z) ) \notin \mathcal{C}_{\alpha }, \quad\text{whenever } \, |\omega | < \tau \text{.}\end{align*} $$

In particular, while $\{z, g(z), \ldots , g^{k-1}(z)\}$ remains in $\mathbb {D}_{\delta }( \mathbf {0} )$ , $g^k (z) \notin \mathcal {C}_{\alpha }$ .

We keep this result aside for now and move on to understand how isotopies of the kind defined in the Extension Lemma act on cones.

Lemma 3.2. Let $( g_t )_{t \in \mathbb {I}}$ be a planar diffeotopy such that the origin is a fixed point and $\mathrm {D} g_t ( \mathbf {0} )$ has the x-axis as an invariant direction for every t. Then, given $0 < \alpha < \pi /2$ , there exist $0 < \beta < \alpha $ and $\rho> 0$ such that $z \in \mathbb {D}_{\rho }( \mathbf {0} ) \setminus \mathcal {C}_{\alpha }$ imply $g_t(z) \notin \mathcal {C}_{\beta }$ for every $t \in \mathbb {I}$ .

Proof. For each fixed $t \in \mathbb {I}$ , let $A_t :=\mathrm {D} g_t ( \mathbf {0} )$ . Consider $v_{\alpha }$ and $v_{\alpha }^{*}$ unit vectors of polar angle α and $2\pi - \alpha $ , respectively, whose spans delimit $\mathcal {C}_{\alpha }$ . Upon defining $\beta _t = \min \lbrace \alpha , {\mathrm {ang}}( A_t v_{\alpha }, {}^{\partial }{\hspace {-5.55542pt}\; / \, \hspace {-4.44443pt}}_{\partial x} ) , {\mathrm {ang}}( A_t v_{\alpha }^{*}, {}^{\partial }{\hspace {-5.55542pt}\; / \, \hspace {-4.44443pt}}_{\partial x} ) \rbrace $ , we have $0 < \beta _t \leq \alpha $ and $A_t v \notin \mathcal {C}_{\beta _t}$ whenever $v \notin \mathcal {C}_{\alpha }$ . This yields a global solution to the associated linear problem.

Once this is done, let $\varepsilon _t :=r_t \, c_t \, \sin ( {\beta _t}/{4} )$ , where $r_t = ( 1 + \sin ( \beta _t / 4 ) )^{-1}$ and $c_t> 0$ is such that $|A_t v | \geq c_t |v |$ for every v. One then obtains $\rho _t> 0$ such that:

(2) $$ \begin{align} |g_t (z) - A_t z | \leq {\varepsilon _t} |z | / 2 \quad \text{whenever} \, 0 < |z | < \rho _t \text{.}\end{align} $$

With respect to the C ¹ topology (see e.g. [1]), continuity of the isotopy yields $\delta _t> 0$ such that $s \in \mathbb {I}$ and $|s - t | < \delta _t$ imply:

(3) $$ \begin{align} \sup \nolimits _{\, \overline{\mathbb{D}}_{\rho _t}( \mathbf{0} )} \big \{ | g_s - g_t | , \Vert \mathrm{D}g_s - \mathrm{D}g_t \Vert \big \} \leq {\varepsilon _t}/{2} \text{.}\end{align} $$

Thus, when $|s - t | < \delta _t$ and $|z | < \rho _t$ simultaneously, we have $| g_s (z) - A_t z | \leq \varepsilon _t |z |$ , by the mean value inequality along with equations (2) and (3).

It follows that $|g_s (z) | \geq ( c_t - \varepsilon _t ) |z |$ , and the right-hand side of the inequality is strictly positive. This allows us to conclude that $g_s(z)$ lies in a closed disk centered at A _t z and of radius $\varepsilon _t |z |$ , to which the origin is an external point. In particular, ${\mathrm {ang}}( g_s (z), A_t z ) < \pi / 2$ . An elementary chord length formula then implies

$$ \begin{align*}\sin \bigg[ \frac{ {\mathrm{ang}}( g_s(z), A_t z ) }{2} \bigg] \leq \frac{ | g_s (z) - A_t z | }{ ( c_t - \varepsilon _t ) |z | } \leq \frac{ \varepsilon _t |z | }{ ( c_t - \varepsilon _t ) |z | } = \sin \bigg( \frac{\beta _t}{4} \bigg) \text{.}\end{align*} $$

We conclude that ${\mathrm {ang}}( g_s(z), A_t z ) \leq \beta _t / 2$ , as long as $|s-t | < \delta _t$ and $0< |z | < \rho _t$ . If also $z \notin \mathcal {C}_{\alpha }$ , the linear case implies $A_t z \notin \mathcal {C}_{\beta _t}$ , and $g_s (z) \notin \mathcal {C}_{\beta _t / 2}$ follows. Covering 𝕀 by finitely many $( t - \delta _t, t + \delta _t )$ intervals finishes the proof.

Corollary 3.3. Let $( g_t )_{t \in \mathbb {I}}$ be a planar diffeotopy such that the origin is a fixed point and $\mathrm {D} g_t ( \mathbf {0} )$ has the x-axis as an invariant direction for every t. Then, given $0 < \alpha < \pi /2$ , there exist $0 < \beta ^{-} < \beta ^{+} < \alpha $ and $\rho> 0$ such that:

1. (1) $z \in \mathbb {D}_{\rho }( \mathbf {0} )$ and $z \notin \mathcal {C}_{\alpha }$ imply $g_t(z) \notin \mathcal {C}_{\beta ^{+}}$ for every $t \in \mathbb {I}$ ;

2. (2) $z \in \mathbb {D}_{\rho }( \mathbf {0} )$ and $z \in \mathcal {C}_{\beta ^{-}}$ imply $g_t(z) \in \mathcal {C}_{\beta ^{+}}$ for every $t \in \mathbb {I}$ .

Proof. Given α, Lemma 3.2 yields $0 < \beta ^{+} < \alpha $ and $\rho ^{+}> 0$ such that $g_t(z) \notin \mathcal {C}_{\beta ^{+}}$ for every $t \in \mathbb {I}$ , whenever $z \in \mathbb {D}_{\rho ^{+}}( \mathbf {0} )$ and $z \notin \mathcal {C}_{\alpha }$ . The diffeotopy $( g_{t}^{-1} )_{t \in \mathbb {I}}$ also satisfies the hypotheses listed in the lemma. So, for this diffeotopy and the angle $\beta ^{+}$ just encountered, we obtain $\rho ^{-}> 0$ and $0 < \beta ^{-} < \beta ^{+}$ such that $w \in \mathbb {D}_{\rho ^{-}}( \mathbf {0} ) \setminus \mathcal {C}_{\beta ^{+}}$ imply $g_{t}^{-1} (w) \notin \mathcal {C}_{\beta ^{-}}$ for every $t \in \mathbb {I}$ .

Let $\eta> 0$ be such that $|g_t (z) | < \rho ^{-}$ for every $t \in \mathbb {I}$ , whenever $|z | < \eta $ . Then, by setting ρ as $\min \{ \eta , \rho ^{+} \}$ , we have the proposed statements satisfied. Indeed, if that was not the case, $g_s (z) \notin \mathcal {C}_{\beta ^{+}}$ for some $s \in \mathbb {I}$ and $z \in \mathbb {D}_{\rho }( \mathbf {0} ) \cap \mathcal {C}_{\beta ^{-}}$ would lead to a contradiction.

Definition 3.4. Given finite and non-zero points $z,w$ , we let $M [ z,w ] \in {\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{2} (\mathbb {S}^2)$ be the unique Möbius transformation fixing the poles and mapping z to w. Also, we denote $\hat {M} [ z ] :=M [ z,\mathbf {1} ]$ .

We can write down explicit formulas for these transformations and see that $( z, w ) \mapsto M [ z,w ]$ is continuous. Also, if $\mathcal {K} \subset \mathbb {S}^2$ is a non-empty compact set bounded away from 0,

(4) $$ \begin{align}\text{the sets } \hat{M} [ x ] ( \mathcal{K} ) \text{ converge to } \lbrace \mathbf{\infty }\rbrace \text{ on the Hausdorff distance as } x \rightarrow \mathbf{0}\text{.}\end{align} $$

Having settled the notation and technical results, we are now able to prove our Fundamental Lemma.

Lemma 3.5. (Fundamental Lemma)

Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, for each given point $z_0 \notin \Gamma $ , there exists an $\mathcal {I}G_3$ -isotopy $( I^{z_0}_{t} )_{t \geq 0}$ , depending on z ₀ , such that:

1. (1) the trajectory of z ₀ under $I^{z_0}$ does not intersect Γ; and

2. (2) the ω-limit of z ₀ satisfies $\omega _{ I^{z_0} } ( z_0 ) = \lbrace \mathbf {\infty } \rbrace $ .

Proof. Given $z_0 \notin \Gamma $ , we assume for concreteness that it lies on the upper half-plane, and is thus given in polar coordinates as $z_0 = R_0 \, \mathrm {e}^{\mathrm {i} \,\theta _0}$ , $0 < \theta _0 < \pi $ .

Let $( g_t )_{t \in \mathbb {I}}$ be as in the Extension Lemma. Since ∞ is fixed throughout, it can be thought of as a planar $\mathcal {I}G_2$ -diffeotopy such that $\mathrm {D} g_t ( \mathbf {0} )$ has the x-axis as an invariant direction for every t and $\mathrm {D} g_1 ( \mathbf {0} ) = {\mathrm {diag}} [ \unicode{x3bb} , \unicode{x3bb} ^{-1} ]$ , $0 < \unicode{x3bb} < 1$ . To ease notation, we write $g_1 :=g$ and $\mathrm {D} g ( \mathbf {0} ) :=A$ .

Fix some $0 < \alpha < \pi / 2$ such that the direction through θ ₀ is external to $\mathcal {C}_{2 \alpha }$ . With respect to α , let $\delta> 0$ and $\tau> 0$ be as described in Lemma 3.1. Regarding this same α, and also the diffeotopy $( g_t )_{t \in \mathbb {I}}$ , Corollary 3.3 yields a radius $\rho> 0$ and angles $0 < \beta ^{-} < \beta ^{+} < \alpha $ such that, for $z \in \mathbb {D}_{\rho }( \mathbf {0} )$ and every $t \in \mathbb {I}$ , $z \notin \mathcal {C}_{\alpha }$ implies $g_t(z) \notin \mathcal {C}_{\beta ^{+}}$ , while $z \in \mathcal {C}_{\beta ^{-}}$ implies $g_t(z) \in \mathcal {C}_{\beta ^{+}}$ .

Lastly, we characterize the stable manifold W^s of g at 0. Since the stable direction of A is the x-axis, W^s may be assumed to be a Lipschitz graph of the form $y = y(x)$ having horizontal tangent at the origin. So, for τ and $\beta ^{-}$ as obtained above and a sufficiently small radius $\sigma> 0$ , it can be assumed that $\mathrm {W}^{\mathrm {s}}\cap \mathbb {D}_{\sigma }( \mathbf {0} ) \subset \mathcal {C}_{1/2 \min \lbrace \tau , \beta ^{-} \rbrace }$ .

Picking $0 < \rho _{0} < \min \lbrace \delta , \rho , \sigma , 1 \rbrace $ , all of the conditions described so far are satisfied in the disk $\mathbb {D}_{0} :=\mathbb {D}_{\rho _{0}}( \mathbf {0} )$ . Once these choices are made, fix a positive real number $0 < r_1 < \min \lbrace \rho _{0} , \rho _{0}/ R_0 \rbrace $ and let $\rho _1 :=r_1 R_0$ , $\mathcal {K}_0 :=\overline { \partial \mathbb {D}_{\rho _1}( \mathbf {0} ) \setminus \mathcal {C}_{\alpha } }$ .

Then, $\mathcal {K}_0$ is a compact set contained in $\mathbb {D}_{0}$ while not intersecting the stable manifold, so there exists $n_0 \in \mathbb {N}$ such that:

(5) $$ \begin{align}\text{if } n_x = \min \lbrace n \in \mathbb{N} : g^n (x) \notin \overline{\mathbb{D}_{0}} \rbrace \quad\text{then } \, n_x \leq n_0 \text{ for every } x \in \mathcal{K}_0 \text{.}\end{align} $$

Notice that $r_1 < \rho _{0}$ . Since the stable manifold W^s is locally given as a Lipschitz graph $y = y(x)$ , for some τ ₁ with $|\tau _1 | \leq \tfrac 12 \min \lbrace \tau , \beta ^{-} \rbrace $ , we have $r_1 \mathrm {e}^{\mathrm {i} \,\tau _1} \in \mathrm {W}^{\mathrm {s}}_{}$ .

Claim 3.5.1. Define $M_1 (z) :=r_1 \mathrm {e}^{\mathrm {i} \,\tau _1} z$ . Then, M ₁ has the following properties:

1. (i) $M_1 \in {\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{2} (\mathbb {S}^2)$ ;

2. (ii) $v_1 :=M_1 ( \mathbf {1} ) \in \mathrm {W}^{\mathrm {s}}_{}$ ;

3. (iii) $w_1 :=M_1 (z_0) \in \mathcal {K}_0$ .

Proof of claim.

Item (i) follows from the form of M ₁ and item (ii) from the choice of τ ₁. As for item (iii), consider $w_1 = \mathrm {e}^{\mathrm {i} \,\tau _1} (r_1 z_0)$ . Since $r_1 z_0$ also has polar angle θ ₀, w ₁ has polar angle $\theta _0 + \tau _1$ . Then, $|\tau _1 | < \beta ^{-} < \alpha $ and $2 \alpha < \theta _0 < \pi - 2 \alpha $ together imply $\alpha < \theta _0 + \tau _1 < \pi - \alpha $ , so $w_1 \notin \mathcal {C}_{\alpha }$ . Furthermore, $|w_1 | = \rho _1$ , yielding $w_1 \in \mathcal {K}_0$ .

Claim 3.5.2. Let $n_1 :=n_{w_1}$ be as in equation (5). Consider $f_t :=g_{t - \lfloor t \rfloor } \circ g^{\lfloor t \rfloor } \circ M_1$ , where $t \in ( 0, n_1 ] $ . Then, f is an isotopy such that $f_t (z_0) \notin \mathcal {C}_{\beta ^{+}}$ and $f_t (\mathbf {1}) \in \mathcal {C}_{\beta ^{+}}$ for any t.

Proof of claim.

On the one hand, $f_t (z_0) = g_{t - \lfloor t \rfloor } \, ( g^{\lfloor t \rfloor } ( w_1 ) )$ . Since $w_1 \notin \mathcal {C}_{\alpha }$ , each $g^{\lfloor t \rfloor } ( w_1 )$ does not belong to $\mathcal {C}_{\alpha }$ either, as observed in Lemma 3.1. However, it does belong to $\mathbb {D}_{0}$ while $t < n_1$ . Thus, $g_s ( g^{\lfloor t \rfloor } ( w_1 ) ) \notin \mathcal {C}_{\beta ^{+}}$ for every $s = t - \lfloor t \rfloor \in \mathbb {I}$ . On the other hand, $f_t ( \mathbf {1} ) = g_{t - \lfloor t \rfloor } \, ( g^{\lfloor t \rfloor } ( v_1 ) )$ . Since $v_1 \in \mathrm {W}^{\mathrm {s}}_{}$ , each $g^{\lfloor t \rfloor } ( v_1 )$ belongs to $\mathrm {W}^{\mathrm {s}}\cap \mathbb {D}_{0} \subset \mathcal {C}_{\beta ^{-}} \cap \mathbb {D}_{0}$ . Therefore, $g_s ( g^{\lfloor t \rfloor } ( v_1 ) ) \in \mathcal {C}_{\beta ^{+}}$ for every $s = t - \lfloor t \rfloor \in \mathbb {I}$ , as claimed.

The above setting is pictured in Figure 1, where the points

$$ \begin{align*}z_1 :=f_{n_1} (z_0) = g^{n_1} (w_1) \notin \mathcal{C}_{\alpha } \quad \text{and} \quad u_1 :=f_{n_1} (\mathbf{1}) = g^{n_1} (v_1) \in \mathrm{W}^{\mathrm{s}}_{}\end{align*} $$

are introduced.

Figure 1 Promotion by f _t of two parallel processes: points in $\mathcal {K}_0$ are successively dragged out of $\mathbb {D}_{0}$ without entering the $\beta ^{+}$ -cone, while the images of 1 are dragged toward the origin over the stable manifold.

By the choice of n ₁, $|z_1|> \rho _{0}$ . Let $r_2 = \rho _1 / |z_1|$ . Then, $r_2 z_1 \in \mathcal {K}_0$ and $r_2 u_1 \in \mathbb {D}_{0} \cap \mathcal {C}_{\min \lbrace \tau , \beta ^{-} \rbrace }$ . In particular, $\mathrm {e}^{\mathrm {i} \,\tau _2} ( r_2 u_1 ) \in \mathrm {W}^{\mathrm {s}}_{}$ for some τ ₂ such that $|\tau _2 | \leq 2 ( \tfrac 12 \min \lbrace \tau , \beta ^{-} \rbrace ) = \min \lbrace \tau , \beta ^{-} \rbrace $ .

Claim 3.5.3. Define $M_2 (z) :=r_2 \mathrm {e}^{\mathrm {i} \,\tau _2} z$ . Then, M ₂ has the following properties:

1. (i) $M_2 \in {\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{2} (\mathbb {S}^2)$ ;

2. (ii) $v_2 :=M_2 (u_1) \in \mathrm {W}^{\mathrm {s}}_{}$ ;

3. (iii) $w_2 :=M_2 (z_1) \in \mathcal {K}_0$ .

Proof of claim.

Item (i) follows from the form of M ₁ and item (ii) from the choice of τ ₂. As for item (iii), notice that $r_2 \, z_1$ does not belong to $\mathcal {C}_{\alpha }$ —because it is obtained from z ₁ through an homothety—but it does belong to $\mathbb {D}_{0}$ . So Lemma 3.1 implies

$$ \begin{align*}\alpha < \theta ( r_2 z_1 ) - |\tau _2 | \leq \underbrace{\theta ( r_2 z_1 ) + \tau _2}_{= \theta ( w_2 )} \leq \theta ( r_2 z_1 ) + |\tau _2 | < \pi - \alpha \text{,}\end{align*} $$

where it was used that $|\tau _2 | \leq \tau $ . This allows one to conclude that $w_2 \notin \mathcal {C}_{\alpha }$ . Furthermore, $|w_2 | = |r_2 z_1 | = \rho _1$ , establishing that $w_2 \in \mathcal {K}_0$ .

In particular, $n_2 :=n_{w_2}$ is well defined. For $n_1 < t \leq n_1 + n_2$ , consider now the expression $f_t = g_{t - \lfloor t \rfloor } \circ g^{\lfloor t \rfloor - n_1} \circ M_2 \circ f_{n_1}$ . Arguments analogous to the previous ones imply $t \mapsto f_t$ to be continuous over the interval $( n_1, n_1 + n_2 ] $ . Also, $f_t ( z_0 ) \notin \mathcal {C}_{\beta ^{+}}$ and $f_t ( \mathbf {1} ) \in \mathcal {C}_{\beta ^{+}}$ for every t. In a similar fashion, we define inductively for $k \geq 0$ :

(6) $$ \begin{align}f_t :={\begin{cases} \mathrm{id}& \text{if } t = 0 \text{,} \\ g_{t - \lfloor t \rfloor } \circ g^{\lfloor t \rfloor - N_k} \circ M_{k+1} \circ f_{N_k} & \text{on the interval } N_k < t \leq N_{k+1} \text{,} \end{cases}}\end{align} $$

where $N_0 = 0$ , $N_k = \sum _{i=1}^{k} n_i$ , and the numbers n _k and maps M _k satisfy:

• • $M_{k+1} \in {\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{2} (\mathbb {S}^2) $ is a transformation of the form

  $$ \begin{align*}M_{k+1} \, (z) = r_{k+1} \, \mathrm{e}^{\mathrm{i} \,\tau _{k+1}} z \text{,}\end{align*} $$
  mapping $z_k :=f_{N_k} (z_0) \notin \overline {\mathbb {D}_{0}}$ to a point $w_{k+1} \in \mathcal {K}_0$ and $u_k :=f_{N_k} (\mathbf {1}) \in \mathrm {W}^{\mathrm {s}}_{}$ to a point $v_{k+1} \in \mathrm {W}^{\mathrm {s}}_{}$ , via an homothety of scaling factor $r_{k+1} = \rho _1 / |z_k| < 1$ and a rotation of angle $|\tau _{k+1} | \leq \min \{\tau , \beta ^{-}\}$ ;

• • $n_k :=n_{w_k} \leq n_0$ is given as in equation (5).

The following properties hold, by construction:

1. (i) $f_t \in G_2$ for every $t \geq 0$ ;

2. (ii) $t \mapsto f_t$ is continuous over each interval of the form $( N_k, N_{k+1} ] $ ;

3. (iii) $f_t ( z_0 ) \notin \mathcal {C}_{\beta ^{+}}$ and $f_t ( \mathbf {1} ) \in \mathcal {C}_{\beta ^{+}}$ for every $t \geq 0$ .

Claim 3.5.4. For $t \geq 0$ , let

$$ \begin{align*}I_{t}^{z_0} :=\hat{M} [f_{t} ( \mathbf{1} )] \circ f_t \text{,}\end{align*} $$

where $\hat {M} [ \cdot ]$ is as in Definition 3.4. Then, $( I^{z_0}_{t} )_{t \geq 0}$ is an $\mathcal {I}G_3$ -isotopy.

Proof of claim.

Since $f_t \in G_2$ for every $t \geq 0$ , it is clear that $I^{z_0}_{t} \in G_3$ for every $t \geq 0$ . One must see that $t \mapsto I^{z_0}_{t}$ defines a continuous curve of homeomorphisms. It is, a priori, as continuous as $t \mapsto f_t$ . Thus, all that is needed to check is continuity from the right at the left endpoints of each interval $( N_k, N_{k+1} ] $ . For $0 < h < 1$ ,

$$ \begin{align*}I^{z_0}_{N_k + h} = \hat{M} [f_{N_k + h} ( \mathbf{1} )] \circ f_{N_k + h} = \hat{M} [g_{h} ( v_{k+1} )] \circ g_{h} \circ M_{k+1} \circ f_{N_k} \text{.}\end{align*} $$

However, notice that

$$ \begin{align*}\hat{M} [ u_k ] \circ M_{k+1}^{-1} \circ M [g_h ( v_{k+1} ) , v_{k+1}]\end{align*} $$

is a Möbius transformation fixing the poles and mapping $g_h ( v_{k+1} )$ to 1. By sharp 3-transitivity, it must actually be $\hat {M} [ g_{h} ( v_{k+1} ) ]$ . However, since $g_{h} \rightarrow \mathrm {id}$ as $h \rightarrow 0^{+}$ , the above expression implies $\hat {M} [ g_{h} ( v_{k+1} ) ] \rightarrow \hat {M} [ u_{k} ] \circ M_{k+1}^{-1}$ as $h \rightarrow 0^{+}$ . Consequently,

$$ \begin{align*}I^{z_0}_{N_k + h} \rightarrow \hat{M} [ u_k ] \circ M_{k+1}^{-1} \circ g_{0} \circ M_{k+1} \circ \, f_{N_k} &= \hat{M} [ u_k ] \circ f_{N_k} \\ &= \hat{M} [f_{N_k} ( \mathbf{1} )] \circ f_{N_k} = I^{z_0}_{N_k} \text{.}\\[-3pc] \end{align*} $$

Claim 3.5.5. $\gamma _{I^{z_0}} ( z_0 ) \cap \Gamma = \emptyset \text {.}$

Proof of claim.

Since $f_t ( \mathbf {1} ) \in \mathcal {C}_{\beta ^{+}}$ for every $t \geq 0$ , the transformation $\hat {M} [f_t ( \mathbf {1} )]$ may be explicitly written as $\hat {M} [f_t ( \mathbf {1} )] (z) = |f_t ( \mathbf {1} ) |^{-1} \, \mathrm {e}^{ \mathrm {i} \,\psi } z$ , where $|\psi | \leq \beta ^{+}$ . As we also know that $f_{t} (z_0) \notin \mathcal {C}_{\beta ^{+}}$ for every $t \geq 0$ , $\hat {M} [f_{t} ( \mathbf {1} )] \circ f_t (z_0)$ remains on the upper half-plane without ever touching the x-axis. This establishes (1).

Claim 3.5.6. $f_t ( \mathbf {1} ) \rightarrow \mathbf {0}$ as $t \rightarrow + \infty $ .

Proof of claim.

Further shrinking $\mathbb {D}_{0}$ if necessary, we may assume that $|g(z) | < |z |$ for every $z \in \mathrm {W}^{\mathrm {s}}_{}$ . Then, since $|u_{k}| = | g^{n_{k}} ( v_{k} )| < |v_{k} |$ and $|v_{k+1} | = r_{k+1} |u_k |$ , we have

(7) $$ \begin{align}|v_{k+1} | < ( {\rho _1}/{\rho _0} )^{k} \text{,}\end{align} $$

for $r_{k+1}=\rho _1 / |z_k| \leq \rho _1 / \rho _0$ and $v_1 \in \mathbb {D}_{0}$ has a norm not greater than one.

Now, on each interval $( N_k, N_{k+1} ] $ , the expression $f_t ( \mathbf {1} ) = g_{t - \lfloor t \rfloor } \circ g^{\lfloor t \rfloor - N_{k}} ( v_{k+1} )$ holds. As t ranges through this interval, the quantity $t - \lfloor t \rfloor $ ranges over the interval $[0, 1]$ and the quantity $\lfloor t \rfloor - N_k$ ranges through $\lbrace 0, \ldots , n_{k+1} \rbrace \subset \lbrace 0, \ldots , n_0 \rbrace $ . Thus,

(8) $$ \begin{align}\lbrace f_t ( \mathbf{1} ) : N_k < t \leq N_{k+1} \rbrace \subset \lbrace g_s ( g^{i} ( v_{k+1} ) ) : s \in \mathbb{I} \text{ and } 1 \leq i \leq n_0 \rbrace \text{.}\end{align} $$

By the condition on the stable manifold, $|g^i ( v_{k+1} ) | < |v_{k+1} |$ for each $1 \leq i \leq n_0$ . The inclusion above—along with equation (7)—thus implies:

(9) $$ \begin{align}\lbrace f_t ( \mathbf{1} ) : N_k < t \leq N_{k+1} \rbrace \subset \lbrace g_s (z) : s \in \mathbb{I} \text{ and } z \in \mathbb{D}_{( \rho _1 / \rho _0 )^k} ( \mathbf{0} ) \rbrace \text{.}\end{align} $$

Since $( \rho _1 / \rho _0 ) < 1$ , the sets at the right-hand side of equation (9) are nested and decreasing with $k \in \mathbb {N}$ . In particular,

(10) $$ \begin{align}\sup \lbrace |f_t (\mathbf{1}) | : t> N_k \rbrace \leq \max \lbrace | g_s (z) | : s \in \mathbb{I} \text{ and } z \in \mathbb{D}_{( \rho _1 / \rho _0 )^k} ( \mathbf{0} ) \rbrace \text{.}\end{align} $$

Lastly, let $\varepsilon> 0$ be given and fix $\eta> 0$ such that $|z | < \eta $ implies $|{g_{s} ( z )}| < \varepsilon $ for every $s \in \mathbb {I}$ . Then, if $k_0 \in \mathbb {N}$ is so large that $(\rho _1 / \rho _{0})^{k_0} < \eta $ , equation (10) yields $|f_t ( \mathbf {1} )| < \varepsilon $ whenever $t> N_{k_0}$ .

We may now finish the proof. We know that $f_t (z_0) = g_{t - \lfloor t \rfloor } \circ g^{\lfloor t \rfloor - N_{k}} ( w_{k+1} )$ on each interval $( N_k, N_{k+1} ] $ , where $w_k \in \mathcal {K}_0$ for every $k \in \mathbb {N}$ . Observing the same ranges as in the proof of the previous claim, we see that, for every $t \geq 0$ ,

$$ \begin{align*}f_t ( z_0 ) \in \lbrace g_{s} \circ g^i ( z ) : s \in \mathbb{I}, 1 \leq i \leq n_0 \, \text{ and } \, z \in \mathcal{K}_0 \rbrace :=\mathcal{K} \text{.}\end{align*} $$

However, $\mathcal {K}$ is a compact set bounded away from 0. Since $f_t ( \mathbf {1} ) \rightarrow \mathbf {0}$ as $t \rightarrow + \infty $ and $I^{z_0} (z_0) \in \hat {M} [f_t ( \mathbf {1} )] ( \mathcal {K} )$ , the remark in equation (4) implies (2).

Corollary 3.6. Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, for each given point $z_0 \notin \Gamma $ and each pair of points $a, b \in \lbrace \mathbf {0},\mathbf {1},\mathbf {\infty } \rbrace $ , there exists a full-time $\mathcal {I}G_3$ -isotopy $( \, I^{z_0}_{ab} (t ,\cdot ) \, )_{t \in \mathbb {R}}$ such that:

1. (1) the trajectory of z ₀ under $I^{z_0}_{ab}$ does not intersect Γ;

2. (2) the α and ω limits of z ₀ satisfy $\alpha _{ I^{z_0}_{ab} } ( z_0 ) = \lbrace a \rbrace $ and $\omega _{ I^{z_0}_{ab} } ( z_0 ) = \lbrace b \rbrace $ .

Proof. Let $z_0 \notin \Gamma $ and $a, b \in \lbrace \mathbf {0},\mathbf {1},\mathbf {\infty } \rbrace $ be given. We denote by $T_{a b}$ the (unique) idempotent Möbius transformation permuting a and b and fixing the remaining reference point. Each $T_{a b}$ leaves Γ invariant, so neither $T_{\mathbf {\infty } b} ( z_0 )$ nor $T_{\mathbf {\infty } a} ( z_0 )$ lie on Γ. The Fundamental Lemma then yields $\mathcal {I}G_3$ -isotopies $\big ( \, I^{ T_{\mathbf {\infty } b} (z_0) \, }_{t} \big )_{t \geq 0}$ and $\big ( \, I^{ T_{\mathbf {\infty } a} (z_0) }_{t} \, \big )_{t \geq 0}$ as described therein, from which we define $I^{z_0}_{ab} : \mathbb {R}\times \mathbb {S}^2\rightarrow \mathbb {S}^2$ as

(11) $$ \begin{align}I^{z_0}_{ab} ( t, \, z ) = {\begin{cases} T_{\mathbf{\infty } b} \circ I^{ T_{\mathbf{\infty } b} ( z_0 ) }_{t} \circ T_{\mathbf{\infty } b} \; (z) & \text{if } t \geq 0 \text{,} \\ T_{\mathbf{\infty } a} \circ I^{ T_{\mathbf{\infty } a} ( z_0 ) }_{-t} \circ T_{\mathbf{\infty } a} \; (z) & \text{if } t \leq 0 \text{.} \end{cases}}\end{align} $$

The statements then follow by noting that $t \mapsto -t$ turns the ω into the α-limit.

4 A crossing lemma

Let us to now analyze the behavior of Γ under the action of G ₀. Before doing so, we agree that, given three distinct points $a,b,c$ , we denote by $\hat {M} [ a,b,c ]$ the unique Möbius transformation mapping a to 0, b to 1, and c to ∞. If the points are all finite and non-zero, this map can be explicitly written as a certain cross ratio, from which it is seen that $( a,b,c ) \mapsto \hat {M} [ a,b,c ]$ is continuous.

Lemma 4.1. Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, there exists an $\mathcal {I}G_3$ -isotopy $( h_t )_{t \in \mathbb {I}}$ such that $|\, h_1 ( \Gamma ) \cap \Gamma \, | = 4$ .

Proof. The Extension Lemma yields an $\mathcal {I}G_2$ -isotopy $( g_t )_{t \in \mathbb {I}}$ such that $g :=g_1$ has a hyperbolic saddle fixed point at the origin. Fix any sufficiently small $0< r_0< 1$ such that $|\, S_{r_0} \cap g ( S_{r_0} ) \, | = 4$ , and let $a,b,c,d \in S_{r_0}$ be four consecutive points in the usual anticlockwise cyclic order such that $S_{r_0} \cap g ( S_{r_0} ) = \lbrace g(a), g(b), g(c), g(d) \rbrace $ . Then, we may consider the unique $M_0 \in {\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ mapping the ordered triple $( \mathbf {0},\mathbf {1},\mathbf {\infty } )$ onto $( a,b,c )$ . Clearly, $M_0 = \hat {M} [ a,b,c ]^{-1}$ and $M_0 ( \Gamma ) = S_{r_0}$ . By defining

$$ \begin{align*}h_t :=\hat{M} [ g_t(a),g_t(b),g_t(c) ] \circ g_t \circ M_0 \; \text{,} \quad 0 \leq t \leq 1 \; \text{,}\end{align*} $$

we obtain an $\mathcal {I}G_3$ -isotopy $( h_t )_{t \in \mathbb {I}}$ . Since $\hat {M} [ g(a),g(b),g(c) ] ( S_{r_0} ) = \Gamma $ and Möbius transformations are bijections of the sphere,

$$ \begin{align*}| h_1 ( \Gamma ) \cap \Gamma | &= | \hat{M} [ g(a),g(b),g(c) ] \circ g \circ M_0 \, ( \Gamma ) \, \cap \, \hat{M} [ g(a),g(b),g(c) ] ( S_{r_0} ) | \\ &= | \hat{M} [ g(a),g(b),g(c) ] \circ g ( M_0 (\Gamma ) ) \, \cap \, \hat{M} [ g(a),g(b),g(c) ] ( S_{r_0} ) | \\ &= | \hat{M} [ g(a),g(b),g(c) ] \circ g ( S_{r_0} ) \, \cap \, \hat{M} [ g(a),g(b),g(c) ] ( S_{r_0} ) | \\ &= |\, S_{r_0} \cap g ( S_{r_0} ) \, | = 4 \text{.}\\[-3pc] \end{align*} $$

Corollary 4.2. Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, there exist $z_0 \in \Gamma \setminus \lbrace \mathbf {0},\mathbf {1},\mathbf {\infty } \rbrace $ and an $\mathcal {I}G_3$ -isotopy $( h_t )_{t \in \mathbb {I}}$ such that $h_1 ( z_0 ) \notin \Gamma $ .

Proof. In the language of Lemma 4.1, let $h_1 ( \Gamma ) \cap \Gamma = \lbrace \mathbf {0}, \mathbf {1}, \mathbf {\infty }, w_0 \rbrace $ . It suffices to take any $z_0 \in \Gamma \setminus \lbrace \mathbf {0}, \mathbf {1}, \mathbf {\infty }, h_1^{-1} (w_0) \rbrace $ .

Finite points $a,b$ on the meridian Γ are identified with their real counterparts on the x-axis. This induces a natural ordering, for which one may speak of the oriented interval with endpoints $a,b$ . Whenever $a < b$ with respect to this ordering:

• • $[ a, b ]$ denotes the arc of the meridian Γ with endpoints $a,b$ and not containing ∞, corresponding to the compact interval of the x-axis with the associated endpoints;

• • $[ b, a ]$ denotes the arc of the meridian Γ with endpoints $a,b$ and containing ∞, which projects onto $( -\infty , a ] \cup [ b, + \infty ) $ .

If $b = \mathbf {\infty }$ , then the corresponding arcs are defined via stereographic projection as $[ \mathbf {\infty }, a ] = ( - \infty , a ] \cup \lbrace \mathbf {\infty } \rbrace $ and $[ a, \mathbf {\infty } ] = [ a, + \infty ) \cup \lbrace \mathbf {\infty } \rbrace $ . Lastly, open and half-open arcs of Γ are defined accordingly by deletion of the suitable endpoints from the corresponding closed arcs.

Lemma 4.3. (Crossing Lemma)

Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Then, there exist a point $\hat {z}$ on the open segment $( \mathbf {0}, \mathbf {1} )$ of Γ and an $\mathcal {I}G_3$ -isotopy $( J_t )_{t \in [ -1,1 ]}$ such that:

1. (1) the trajectory of $\hat {z}$ under J only intersects Γ on the arc $( \mathbf {0}, \mathbf {1} )$ ;

2. (2) $J_{-1} ( \hat {z} ) \in \mathcal {H}^{-}$ and $\, J_{\, 1} ( \hat {z} ) \in \mathcal {H}^{+}$ .

Proof. By Corollary 4.2, we may fix an $\mathcal {I}G_3$ -isotopy $( h_t )_{t \in \mathbb {I}}$ and a point $z_0 \in \Gamma $ such that $h_1 ( z_0 ) \notin \Gamma $ . Without loss of generality, assume $h_1 ( z_0 ) \in \mathcal {H}^{+}$ and choose $u_0 \in \mathcal {H}^{-}$ near z ₀ such that $h_1 ( u_0 ) \in \mathcal {H}^{+}$ as well. Consider the continuous path defined as $\gamma (t) = h_t ( u_0 )$ . The sets $\gamma ^{-1} ( \mathcal {H}^{-} )$ , $\gamma ^{-1} ( \Gamma )$ , and $\gamma ^{-1} ( \mathcal {H}^{+} )$ form a partition of 𝕀. Let

$$ \begin{align*}t^{-} = \sup \lbrace t \in \mathbb{I} : \gamma (t) \in \mathcal{H}^{-} \rbrace \quad \text{and} \quad t^{+} = \inf \lbrace t \in [ t^{-} , 1 ] : \gamma (t) \in \mathcal{H}^{+} \rbrace \text{.}\end{align*} $$

We have $0 < t^{-} \leq t^{+} < 1$ and $[ t^{-} , t^{+} ] \subset \gamma ^{-1} ( \Gamma )$ . Since γ cannot meet the points $\{ \mathbf {0},\mathbf {1},\mathbf {\infty }\}$ , $\gamma ( [ t^{-} , t^{+} ] )$ is a compact arc contained within one of the three connected components of $\Gamma \setminus \{\mathbf {0}, \mathbf {1}, \mathbf {\infty }\}$ , being at a positive distance $\rho> 0$ from those three points. Let $\delta> 0$ be such that

(12) $$ \begin{align}s \in \mathbb{I}\quad \text{and} \quad |s - t^{\pm }| < \delta \quad \text{imply } d ( \gamma (s), \gamma (t^{\pm }) ) < \rho \text{.}\end{align} $$

Then, we may fix $t_1 \in \gamma ^{-1} ( \mathcal {H}^{-} ) \cap ( t^{-} - \delta , t^{-} )$ and $t_2 \in \gamma ^{-1} ( \mathcal {H}^{+} ) \cap ( t^{+}, t^{+} + \delta )$ . Let $\sigma : [ -1,1 ] \rightarrow [ t_1, t_2 ]$ be any increasing bijection such that $\sigma (0)$ is the midpoint of the interval $[ t^{-} , t^{+} ]$ . Defining $(\tilde {J}_t)_{t \in [-1,1]}$ by $\tilde {J}_t (z) = h_{\sigma (t)} \circ h_{\sigma (0)}^{-1} \, (z)$ yields an $\mathcal {I}G_3$ -isotopy.

Letting $\tilde {z} = \gamma ( \sigma (0) )$ , we have $\tilde {J}_{1} ( \tilde {z} ) \in \mathcal {H}^{+}$ and $\tilde {J}_{-1} ( \tilde {z} ) \in \mathcal {H}^{-}$ . Furthermore, the trajectory of $\tilde {z}$ under $\tilde {J}$ is $\gamma _{\tilde {J}} ( \tilde {z} ) = \gamma ( [ t^{-} - \delta ^{-} , t^{+} + \delta ^{+} ] )$ . In particular, it follows from the choice of ρ and from equation (12) that any point in $\gamma _{\tilde {J}} ( \tilde {z} ) \cap \Gamma $ lies on the same connected component of $\Gamma \setminus \lbrace \mathbf {0}, \mathbf {1}, \mathbf {\infty } \rbrace $ as the segment $\gamma ( [ t^{-} , t^{+} ] )$ , which is precisely the component containing $\tilde {z}$ .

Considering the cyclic order induced in Γ, let $( a, b )$ be the only interval among $( \mathbf {0}, \mathbf {1} )$ , $( \mathbf {1}, \mathbf {\infty } )$ , and $( \mathbf {\infty }, \mathbf {0} )$ in which $\tilde {z}$ lies. Upon defining $\lbrace c \rbrace = \lbrace \mathbf {0},\mathbf {1},\mathbf {\infty } \rbrace \setminus \lbrace a, b \rbrace $ , the Möbius transformation $\hat {M} [ a,b,c ]$ —which can be the identity—preserves this cyclic order, leaving the hemispheres invariant. Thus, we may take $\hat {z} = \hat {M} [ a,b,c ] ( \tilde {z} )$ and $J_t = \hat {M} [ a,b,c ] \circ \tilde {J}_{\, t} \circ \hat {M} [ a,b,c ]^{-1}$ .

5 Closing arguments

5.1 Proof of Theorem A

From now on, a subgroup $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ is fixed throughout, and we let the point $\hat {z} \in \Gamma $ and the $\mathcal {I}G_3$ -isotopy $( J_t )_{t \in [-1,1]}$ be as in the Crossing Lemma. Upon denoting $\hat {z}_{-} :=J_{-1} ( \hat {z} )$ and $\hat {z}_{+} :=J_{1} ( \hat {z} )$ , we define the set

(13) $$ \begin{align}\chi :=\overline{ \gamma _{I^{\hat{z}_{-}}} ( \hat{z}_{-} ) \cup \gamma _{J} ( \hat{z} ) \cup \gamma _{I^{\hat{z}_{+}}} ( \hat{z}_{+} ) } \text{.}\end{align} $$

Above, $I^{\hat {z}_{-}}$ and $I^{\hat {z}_{+}}$ are the isotopies yielded by the Fundamental Lemma when considering the points $\hat {z}_{-} \in \mathcal {H}^{-}$ and $\hat {z}_{+} \in \mathcal {H}^{+}$ .

On the sphere, χ is a continuum, for it is the closure of the union of connected curves with points in common, and consists of adjoining $\lbrace \mathbf {\infty } \rbrace $ to that union, as we suggest in Figure 2.

Figure 2 The set χ is constructed by gluing isotopy trajectories with points in common, one of which is bounded away from ∞ and two of which accumulate at ∞, and then taking their closure.

Claim 5.1.1. The set χ separates $( \mathbf {\infty }, \mathbf {0} )$ and $( \mathbf {1}, \mathbf {\infty } )$ in the following sense: for every path $\alpha : \mathbb {I}\rightarrow \mathbb {S}^2$ such that $\alpha (0) \in ( \mathbf {\infty }, \mathbf {0} )$ and $\alpha (1) \in ( \mathbf {1}, \mathbf {\infty } )$ , we have $\alpha ( \mathbb {I} ) \cap \chi \ne \emptyset $ .

Proof. If $\mathbf {\infty }\in \alpha ( \mathbb {I} )$ , there is nothing to prove. Otherwise, suppose that α never passes through ∞. By a standard Zorn lemma argument, it can also be assumed simple. By the hypothesis made upon the endpoints of α, the following are well defined:

(14) $$ \begin{align}t^{-} :=\sup \lbrace t \in \mathbb{I} : \alpha (t) \in (. \mathbf{\infty }, \mathbf{0} ]. \rbrace \quad \text{and} \quad t^{+} :=\inf \lbrace t \in (. t^{-}, 1 ]. : \alpha (t) \in [. \mathbf{1},\mathbf{\infty } ). \rbrace \text{.}\end{align} $$

Continuity of α implies $\alpha ( t^{-} ) \in ( \mathbf {\infty },\mathbf {0} ] $ and $\alpha ( t^{+} ) \in [ \mathbf {1},\mathbf {\infty } ) $ . In particular, $t^{+}> t^{-}$ , since these points are mapped to disjoint arcs. We further notice that any intersection between $\alpha ( ( t^{-}, t^{+} ) )$ and Γ takes place in the open arc $( \mathbf {0},\mathbf {1} )$ , by the very own definition of $t^{-}$ and $t^{+}$ . On the plane, we define the following continuous map, which is a line—that is, a simple and proper path:

$$ \begin{align*} \ell (t) = \begin{cases} \displaystyle \frac{1-t \hphantom{^{-}} }{1-t^{-}} ( { \alpha (t^{-}) - \hat{z} } ) + \hat{z} & \: \text{if } t \leq t^{-} \text{, } \\[0.25em] \alpha (t) & \: \text{if } t^{-} < t < t^{+} \text{, } \\[0.25em] \displaystyle \frac{1+t \hphantom{^{+}} }{1+t^{+}} ( { \alpha (t^{+}) - \hat{z}} ) +\hat{z} &\: \text{if } t \geq t^{+} \text{.} \end{cases} \end{align*} $$

Indeed, to see that ℓ is simple, notice first that since $\alpha (t^{-}) \in ( \mathbf {\infty },\mathbf {0} ] $ , $\alpha (t^{+}) \in [ \mathbf {1},\mathbf {\infty } ) $ and $\hat {z} \in ( \mathbf {0},\mathbf {1} )$ all lie in disjoint segments, both $( \alpha (t^{-}) - \hat {z} )$ and $( \alpha (t^{+}) - \hat {z} )$ are non-zero. Thus, the restrictions $\ell |_{t \leq t^{-}}$ and $\ell |_{t \geq t^{+}}$ are injective. Furthermore, $\ell |_{( t^{-}, t^{+} )} = \alpha |_{( t^{-}, t^{+} )}$ is injective as well, for α was assumed simple.

Next, suppose for the sake of contradiction that $\ell ( \bar {t} ) = \ell ( \bar {s} )$ for some $\bar {s} \leq t^{-}$ and $t^{-} < \bar {t} < t^{+}$ . Thinking of Γ as the x-axis, we have $\alpha (t^{-}) \leq \mathbf {0} < \hat {z}$ , from which it follows that $\alpha (t^{-}) - \hat {z}$ is negative. However, since $( 1 - \bar {s} ) / ( 1 - t^{-} ) \geq 1$ , it must be the case that $\ell ( \bar {s} ) \leq \alpha ( t^{-} ) \leq \mathbf {0}$ . In particular, $\ell ( \bar {s} ) \in \Gamma $ , so $\ell ( \bar {t} ) \in \Gamma $ as well. However, then

$$ \begin{align*}\ell (\bar{s}) = \ell (\bar{t}) = \alpha (\bar{t})> \mathbf{0} \geq \ell (\bar{s}) \text{,}\end{align*} $$

which is a contradiction. Analogous reasoning shows that $\ell ( \bar {s} ) = \ell ( \bar {t} )$ cannot happen for $\bar {s} \geq t^{+}$ and $t^{-} < \bar {t} < t^{+}$ both holding simultaneously either. Clearly, ℓ is proper, for $|\ell (t) | \rightarrow \mathbf {\infty }$ as $t \rightarrow \pm \infty $ .

Due to celebrated Jordan–Schoenflies theorem, the orientation inherited by ℓ from ℝ, which coincides with the intrinsic orientation of Γ, divides the plane into two open and connected components, its right $\mathsf {R} ( \ell )$ and its left $\mathsf {L} ( \ell )$ , plus their common boundary ℓ [13].

Consider the compact set $\alpha ( [ t^{-}, t^{+} ] )$ , and fix some closed disk $D \subset \mathbb {R}^2$ containing it. Then, it must be the case that $[ \mathbf {0},\mathbf {1} ] \subset D$ . Indeed, since D is convex, it must contain the segment $[ \alpha ( t^{-} ), \alpha ( t^{+} ) ]$ . However, from equation (14), we know that $\alpha ( t^{-} ) \leq \mathbf {0}$ and $\alpha ( t^{+} ) \geq \mathbf {1}$ , which imply $[ \mathbf {0},\mathbf {1} ] \subset [ \alpha ( t^{-} ), \alpha ( t^{+} ) ]$ .

So, if we now consider the open set $\mathcal {O} :=\mathbb {R}^2 \setminus D$ , we see from the expression of ℓ that $\ell \cap \mathcal {O} = \Gamma \cap \mathcal {O}$ . Also, ℓ traverses this intersection with the same orientation as Γ. These imply $\mathsf {L} ( \ell ) \cap \mathcal {O} = \mathsf {L} ( \Gamma ) \cap \mathcal {O} = \mathcal {H}^{+}\cap \mathcal {O}$ and $\mathsf {R} ( \ell ) \cap \mathcal {O} = \mathsf {R} ( \Gamma ) \cap \mathcal {O} = \mathcal {H}^{-}\cap \mathcal {O}$ . From the Fundamental Lemma, $\gamma _{ I^{\hat {z}_{+}} } ( \hat {z}_{+} )$ is fully contained within H⁺ and accumulates at $\lbrace \mathbf {\infty } \rbrace $ . Since $\mathcal {O}$ is a neighborhood of ∞ on the sphere, it follows that $\gamma _{ I^{\hat {z}_{+}} } ( \hat {z}_{+} ) \cap \mathsf {L} ( \ell ) \ne \emptyset $ . Analogously, $\gamma _{ I^{\hat {z}_{-}} } ( \hat {z}_{-} ) \cap \mathsf {R} ( \ell ) \ne \emptyset $ . This translates to $\chi \cap \mathsf {L} ( \ell ) \ne \emptyset $ and $\chi \cap \mathsf {R} ( \ell ) \ne \emptyset $ . Therefore, the line ℓ intersects the continuum χ.

Let $\bar {t} \in \mathbb {R}$ be such that $\ell (\bar {t}) \in \chi $ . Then, it must be the case that $\bar {t} \in ( t^{-}, t^{+} )$ , for the Crossing Lemma and the Fundamental Lemma imply $( \chi \setminus \lbrace \mathbf {\infty } \rbrace ) \cap \Gamma \subset ( \mathbf {0}, \mathbf {1} )$ . However, $t^{-} < \bar {t} < t^{+}$ means $\ell ( \bar {t} ) = \alpha ( \bar {t} )$ , yielding an intersection between $\alpha ( \mathbb {I} )$ and χ, as claimed.

Now, consider the equivalence relation ∼ _{G 3} described in Definition 1.3. Clearly, $\mathcal {A}_{G_3}( \mathbf {0} ) = \lbrace \mathbf {0} \rbrace $ , $\mathcal {A}_{G_3}( \mathbf {1} ) = \lbrace \mathbf {1} \rbrace $ and $\mathcal {A}_{G_3}( \mathbf {\infty } ) = \lbrace \mathbf {\infty } \rbrace $ . Let us show that $z_0 \in \mathcal {A}_{G_3}( \hat {z} )$ for any $z_0 \in \mathbb {S}^2\setminus \lbrace \mathbf {0},\mathbf {1},\mathbf {\infty } \rbrace $ . By Corollary 4.2, it suffices to consider $z_0 \notin \Gamma $ .

Claim 5.1.2. If $z_0 \notin \Gamma $ , then $z_0 \in \mathcal {A}_{G_3}( \hat {z} )$ .

Proof. Let χ be as in equation (13). Fix $r>0$ such that $\overline {\mathbb {D}}_{r} ( \mathbf {0} )$ and $\overline {\mathbb {D}}_{r} ( \mathbf {1} )$ are disjoint from χ, z ₀, ∞, and also from each other. Consider the $\mathcal {I}G_3$ -isotopy $I^{z_0}_{\mathbf {0} \mathbf {1}}$ yielded by Corollary 3.6. We encounter $S < 0$ maximal such that $I^{z_0}_{\mathbf {0}\mathbf {1}} (S, z_0) \in \partial \overline {\mathbb {D}}_{r} ( \mathbf {0} )$ and $T>0$ minimal such that $I^{z_0}_{\mathbf {0}\mathbf {1}} (T, z_0) \in \partial \overline {\mathbb {D}}_{r} ( \mathbf {1} )$ .

Let $\alpha : \mathbb {I}\rightarrow \mathbb {R}^2$ depart from a point $\alpha (0) \in ( \mathbf {\infty }, \mathbf {0} ) \cap \overline {\mathbb {D}}_{r} ( \mathbf {0} )$ ; follow on a straight line until it reaches $\alpha (1/4) = I^{z_0}_{\mathbf {0}\mathbf {1}} (S, z_0)$ at the disk’s boundary; follow z ₀’s isotopy path under $I^{z_0}_{\mathbf {0} \mathbf {1}}$ until it first reaches the boundary of the disk $\overline {\mathbb {D}}_{r} ( \mathbf {1} )$ at $\alpha (3/4) = I^{z_0}_{\mathbf {0}\mathbf {1}} (T, z_0)$ ; then move on a straight line until it reaches a point $\alpha (1) \in ( \mathbf {1}, \mathbf {\infty } ) \cap \overline {\mathbb {D}}_{r} ( \mathbf {1} )$ .

Claim 5.1.1 implies α to intersect $\chi \setminus \lbrace \mathbf {\infty } \rbrace $ . However, since the segments $\alpha ( [ 0, {1}/{4} ] )$ and $\alpha ( [ {3}/{4}, 1 ] )$ are contained within disks disjoint from χ, we must have $\alpha ( \, ( {1}/{4}, {3}/{4} ) \, ) \cap ( \chi \setminus \lbrace \mathbf {\infty } \rbrace ) \ne \emptyset $ . Since by construction $\alpha ( \, ( {1}/{4}, {3}/{4} ) \, ) \subset \gamma _{ I^{z_0}_{\mathbf {0}\mathbf {1}} } ( z_0 )$ , Lemma 1.4 is readily seen to imply $z_0 \in \mathcal {A}_{G}( \hat {z} )$ . This process is conveyed in Figure 3.

Figure 3 After leaving a compact neighborhood of 0 disjoint of χ and before entering a neighborhood of 1 disjoint from χ, the path of z ₀ under $I^{z_0}_{\mathbf {0}\mathbf {1}}$ must cross the continuum χ.

This is enough to derive the arc 4-transitivity of G ₀, for if $(a,b,c,d)$ and $(p,q,r,s)$ are two given lists of distinct points on the sphere, let $z_0 :=\hat {M} [ a,b,c ] ( d )$ and $w_0 :=\hat {M} [ p,q,r ]^{-1} (s)$ . Then, by Claim 5.1.2, both belong to $\mathcal {A}_{G_3}( \hat {z} )$ . This implies some $\mathcal {I}G_3$ -isotopy $( f_t )_{t \in \mathbb {I}}$ to be such that $f_1 ( z_0 ) = w_0$ . Since ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ is path connected, in particular, $\hat {M} [ p,q,r ]^{-1} \circ f_1 \circ \hat {M} [ a,b,c ] \in G_0$ maps $( a,b,c,d )$ onto $( p,q,r,s )$ . So, the arc 4-transitivity definition is satisfied.

5.2 Discussion on Theorem B

Given three distinct points $p_0, p_1, p_2$ on the plane, let us call the wedge of two circles based at p ₀—each of them traversed once with contrary orientations, while leaving p ₁ and p ₂ in opposite components of their complements—a prototypical figure-8.

Given four distinguished points $P = \lbrace p_0, \ldots , p_3 \rbrace $ on the sphere, a closed loop $\alpha : \mathbb {I}\rightarrow \mathbb {S}^2$ based at p ₀ will be referred to as topological figure-8 (relative to P) if, when p ₃ is placed at infinity, α projects onto a path belonging to the same homotopy class in the fundamental group $\pi _1 (\mathbb {R}^2 \setminus \lbrace p_1, p_2 \rbrace; p_0)$ as a prototypical figure-8. If $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ is a group properly extending ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ , our goal is to realize such an object as the trajectory of a point under an isotopy in G.

To do so, consider χ as defined in equation (13). We may construct an $\mathcal {I}G_3$ isotopy $( K_t )_{t\in \mathbb {R}}$ such that $\chi \setminus \lbrace \mathbf {\infty } \rbrace $ is realized as the trajectory of $\hat {z}$ under K. From it, we may further obtain an $\mathcal {I}G_3$ isotopy $L_t :=T_{\mathbf {0} \mathbf {\infty }} \circ K_t \circ T_{\mathbf {0} \mathbf {\infty }}$ and a point $\hat {y} :=T_{\mathbf {0} \mathbf {\infty }} (\hat {z}) \in ( \mathbf {1}, \mathbf {\infty } )$ with:

1. (i) $\omega _{L} ( \hat {y} ) = \alpha _{L} ( \hat {y} ) = \lbrace \mathbf {0} \rbrace $ ;

2. (ii) $\lbrace L_t (\hat {y}) : t \geq 1 \rbrace \subset \mathcal {H}^{-}$ and $\lbrace L_t (\hat {y}) : t \leq -1 \rbrace \subset \mathcal {H}^{+}$ ;

3. (iii) the trajectory $\gamma _{L} ( \hat {y} )$ only intersects Γ on the open arc $( \mathbf {1}, \mathbf {\infty } )$ .

Claim 5.2.1. There exist a point $\hat {w} \in ( \mathbf {0}, \mathbf {1} )$ and an $\mathcal {I}G_3$ isotopy $( \varphi _t )_{t \in \mathbb {I}}$ such that $\gamma _{\varphi } ( \hat {w} ) \cong \xi _1 \, {\mathrm {rel}}\lbrace \mathbf {0}, \mathbf {1}, \mathbf {\infty } \rbrace $ , where ≅ denotes fixed endpoints homotopy, and ξ ₁ is a circle passing through $\hat {w}$ —traversed once clockwise while leaving 1 on its right and both $\mathbf {0}, \mathbf {\infty }$ on its left.

Proof. Let $( L_t )_{t \in \mathbb {R}}$ and $\hat {y}$ be as in the previous discussion, and consider $\unicode{x3bb} (t) :=L_t ( \hat {y} )$ . We may fix

$$ \begin{align*}\tilde{t} :=\min \lbrace t \in \mathbb{R} : \unicode{x3bb} (t) \in \Gamma \rbrace> -1 \text{.}\end{align*} $$

Item (i) above yields $\unicode{x3bb} ( t )$ in the same connected component of $\chi ^{\mathsf {c}}$ as 0 for $t \ll 0$ , while item (iii) yields $\unicode{x3bb} (\tilde {t}) \in ( \mathbf {1}, \mathbf {\infty } )$ . Claim 5.1.1 then allows us to consider $t^{-} :=\max \lbrace t \leq \tilde {t} : \unicode{x3bb} (t) \in \chi \rbrace $ so that, by item (ii), $L_{t^{-}} ( \hat {y} ) \in \chi \cap \mathcal {H}^{+}$ .

Proceeding analogously for times greater than $\max \lbrace t \in \mathbb {R} : \unicode{x3bb} (t) \in \Gamma \rbrace $ , but using the parts of items (i) and (ii) concerning times $\geq 1$ , we obtain $t^{+}> t^{-}$ such that $L_{t^{+}} ( \hat {y} ) \in \chi \cap \mathcal {H}^{-}$ .

This means that $L_{t^{-}} ( \hat {y} ) = K_a ( \hat {z} )$ and $L_{t^{+}} ( \hat {y} ) = K_b ( \hat {z} )$ , for some $a,b \in \mathbb {R}$ . In particular, there exists an intermediate c for which $K_c ( \hat {z} ) \in \Gamma $ . Upon defining $\hat {w} :=K_c ( \hat {z} )$ , we actually know from the Crossing Lemma that $\hat {w} \in ( \mathbf {0}, \mathbf {1} )$ . Lastly, we define the $\mathcal {I}G_3$ -isotopy $( \varphi _t )_{t \in \mathbb {I}}$ as:

$$ \begin{align*}\varphi _t = {\begin{cases} K_{c + 3t (b-c)} \circ K_{c}^{-1} & \text{if } 0 \leq t \leq {1}/{3} \text{,} \\ L_{2t^{-} - t^{+} + 3t ( t^{+} - t^{-} )} \circ L^{-1}_{t^{-}} \circ K_b \circ K_{c}^{-1} & \text{if } {1}/{3} \leq t \leq {2}/{3} \text{,} \\ K_{3a - 2c + 3t ( c - a )} \circ K^{-1}_{a} \circ L_{t^{+}} \circ L_{t^{-}}^{-1} \circ K_b \circ K_{c}^{-1} & \text{if } {2}/{3} \leq t \leq 1 \text{.} \end{cases}}\end{align*} $$

The path $\gamma (s) :=\varphi _s ( \hat {w} )$ describing the trajectory of the point $\hat {w}$ is indeed a closed loop based at $\hat {w}$ . Along γ, we have four points distinguished during the construction:

• • the starting and ending point $\hat {w} = \gamma (0) = \gamma (1) \in ( \mathbf {0}, \mathbf {1} )$ ;

• • the point $\gamma (1/3) = K_b ( \hat {z} ) = L_{t^{-}} ( \hat {y} ) \in \mathcal {H}^{+}$ ;

• • the point $\hat {y} \in ( \mathbf {1}, \mathbf {\infty } )$ , which is of the form $\hat {y} = \gamma (\bar {s})$ , for some $1/3 < \bar {s} < 2/3$ ; and

• • the point $\gamma (2/3) = L_{t^{+}} ( \hat {y} ) = K_a ( \hat {z} ) \in \mathcal {H}^{-}$ .

We also know that ${\gamma }|{\scriptstyle [0,1/3]}$ and ${\gamma }|{\scriptstyle [2/3,1]}$ only intersect Γ at $( \mathbf {0}, \mathbf {1} )$ , and that ${\gamma }|{\scriptstyle [1/3,\bar {s}]}$ and ${\gamma }|{\scriptstyle [\bar {s},2/3]}$ only intersect Γ at $( \mathbf {1}, \mathbf {\infty } )$ . These suffice to show that γ is homotopic on the plane—with fixed basepoint $\hat {w}$ and relative to $\lbrace \mathbf {0},\mathbf {1} \rbrace $ —to a simple closed curve (say, polygonal) turning once clockwise around 1 and leaving 0 outside of it, as suggested by Figure 4. This amounts to the claimed statement.

Figure 4 The trajectory of $\hat {w}$ under φ is homotopic to a clockwise loop turning around 1, but not 0, with fixed endpoint $\hat {w}$ .

We can now complete the construction. If $\hat {w}^{\prime } :=T_{\mathbf {0} \mathbf {1}} ( \hat {w} )$ , Theorem A yields h in G ₃ such that $\hat {w}^{\prime } = h (\hat {w})$ . Let us consider the $\mathcal {I}G_3$ -isotopy

$$ \begin{align*}\psi _t : = h^{-1} \circ T_{\mathbf{0} \mathbf{1}} \circ \varphi _t \circ T_{\mathbf{0} \mathbf{1}} \circ h \text{.}\end{align*} $$

The trajectory of $\hat {w}$ under ψ is described by the curve $\eta (s) = h^{-1} ( T_{\mathbf {0} \mathbf {1}} \circ \gamma (s) )$ , where γ is as in the proof of Claim 5.2.1. Since $T_{\mathbf {0} \mathbf {1}}$ leaves $\lbrace \mathbf {0},\mathbf {1},\mathbf {\infty } \rbrace $ invariant, $T_{\mathbf {0} \mathbf {1}} \circ \gamma \cong T_{\mathbf {0} \mathbf {1}} \circ \xi _1 \, {\mathrm {rel}}\lbrace \mathbf {0}, \mathbf {1}, \mathbf {\infty } \rbrace $ . However, $T_{\mathbf {0} \mathbf {1}} \circ \xi _1$ is also a circle—traversed once clockwise, while leaving 0 on its right and both $\mathbf {1},\mathbf {\infty }$ on its left. Thus, since $h \in G_3$ , $\xi _2 = h^{-1} \circ T_{\mathbf {0} \mathbf {1}} \circ \xi _1$ is a clockwise simple loop based at $\hat {w}$ , leaving 0 on its right and both $\mathbf {1},\mathbf {\infty }$ on its left. Also, $\eta \cong \xi _2 \, {\mathrm {rel}}\lbrace \mathbf {0}, \mathbf {1}, \mathbf {\infty } \rbrace $ . If we now define the $\mathcal {I}G_3$ -isotopy as

$$ \begin{align*} F_t := \begin{cases} \varphi_{1-2t} \circ \varphi_{1}^{-1} & \text{if } 0 \leq t \leq {1}/{2} \text{, } \\ \psi_{2t - 1} \circ \varphi_1^{-1} & \text{if } {1}/{2} \leq t \leq 1 \text{, } \end{cases} \end{align*} $$

then the trajectory of $\hat {w}$ under the isotopy $( F_t )_{t \in \mathbb {I}}$ is a topological figure-8 relative to the set $P = \lbrace \hat {w}, \mathbf {0}, \mathbf {1}, \mathbf {\infty } \rbrace $ , which is fixed by the terminal diffeomorphism $f = F_1$ .

With respect to f, we now evoke the Nielsen–Thurston classification theorem, as presented in [11]. It takes as input the homeomorphism f and the set P, yielding:

• • a homeomorphism $\Phi : \mathbb {S}^2\rightarrow \mathbb {S}^2$ such that f and Φ are isotopic relative to P; and

• • a (possibly empty) system of f-invariant closed simple loops $\alpha _1, \ldots , \alpha _r$ .

Such reducing curves come equipped with disjoint tubular neighborhoods V _i , not intersecting P, such that the connected components of $\mathbb {S}^2\setminus \bigcup _{i=1}^{r} V_i$ group into invariant cycles, restricted to which either Φ is of finite order—meaning some power of it equals the identity—or Φ is pseudo-Anosov relative to P.

Briefly, pseudo-Anosov means that Φ admits a pair of invariant and measured transverse foliations, one of which is expanded with ratio $\beta> 1$ and the other of which is contracted with ratio $\beta ^{-1}$ . Relative to P means that the points in P are kept fixed under Φ, and manifest as one-prong singularities of the foliations. Such maps are known to have strictly positive topological entropy $h_{\text {top}} ( \Phi ) = \log \beta> 0$ .

Upon interpreting points in P as punctures (as implied by [11] and [3, §7.6]), each connected component of $\mathbb {S}^2\setminus \bigcup _{i=1}^{r} V_i$ must have negative Euler characteristic. Thus, if the reducing system of curves is non-empty, $\hat {w}$ must be enclosed by some α _i along with some other reference point $a \in \lbrace \mathbf {0}, \mathbf {1}, \mathbf {\infty } \rbrace $ .

The key to our argument is that, if α is a simple loop enclosing $\hat {w}$ and a, then its iterates $f^m (\alpha )$ cannot be freely homotopic to α relative to P, allowing us to discard the finite-order cases. This is a property known to hold, and which can be intuitively seen by sliding any such loop along the figure-8. A reference discussion in terms of the action of Dehn half-twists may be found in of [2, Ch. 15].

In short, Φ must be a pseudo-Anosov map. However, the behavior of mappings isotopic to pseudo-Anosov homeomorphisms is described in [7]. More specifically, Theorem 2 therein implies the statement of Theorem B, and also $h_{\text {top}} ( f ) \geq h_{\text {top}} ( \Phi )$ . Thus, f must have positive topological entropy as well.

6 A characterization of the conformal group

Before proving Theorem C, we establish two auxiliary lemmas. The first is a property known a priori to be held by the actual ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . The second establishes conformality at the poles in the subgroup G ₂. In what follows, recall that by homogeneous, we mean a subgroup of $\mathrm {Homeo}_{}(\mathbb {S}^2)$ containing ${\mathrm {Rot}}(\mathbb {S}^2)$ .

Lemma 6.1. Let $G \subset \mathrm {Homeo}_{}(\mathbb {S}^2)$ be a sharply 3-transitive homogeneous group. Then, the subgroup G ₂ fixing the poles permutes parallels.

Proof. A map $g \in G_2$ translates to a planar homeomorphism fixing the origin, for which we must prove that circles centered at the origin are mapped onto circles centered at the origin. Given one such circle γ, let $\unicode{x3bb} :=g ( \gamma )$ . Then, λ is a Jordan loop, containing the origin in its interior. If λ is not a circle, it contains distinct points p _m and p _M of minimum and maximum norm, respectively. For each polar angle $0 \leq \theta < 2\pi $ , the semiradius $\mathbf {r}_{\theta } :=\lbrace t \mathrm {e}^{\mathrm {i} \,\theta } : t \geq 0 \rbrace $ intersects λ in a compact set $\unicode{x3bb} _{\theta }$ . For some θ ₀, it must be the case that $|p | < |p_M |$ for every $p \in \unicode{x3bb} _{\theta _0}$ .

Now, if θ _M is such that $p_M \in \unicode{x3bb} _{\theta _M}$ , let R be a planar rotation mapping the semiradius $\mathbf {r}_{\theta _M}$ onto the semiradius $\mathbf {r}_{\theta _0}$ . Then, $R ( p_M ) \in R ( \unicode{x3bb} )$ , but $R ( p_M ) \in {\mathrm {ext}}\unicode{x3bb} $ , where int and ext are used in the Jordan curve theorem sense, as the bounded and unbounded open connected components of $\unicode{x3bb} ^{\mathsf {c}}$ , sharing λ as their common boundary. It cannot be the case that $R ( \unicode{x3bb} )$ is fully contained within ${\mathrm {ext}}\unicode{x3bb} $ , since $|R ( p_m ) | = |p_m |$ . Therefore, we also must have $R ( \unicode{x3bb} ) \cap \overline {{\mathrm {int}}\unicode{x3bb} } \ne \emptyset $ , and $R ( \unicode{x3bb} ) \cap \unicode{x3bb} \ne \emptyset $ follows.

Therefore, we may obtain $p,q \in \gamma $ such that $R (g(p)) = g(q)$ . Since $p,q$ lie on the same circle, there exists a planar rotation U such that $p = U (q)$ . However then, $g^{-1} \circ R \circ g \circ U$ defines an element of G ₂ fixing q. By sharp 3-transitivity, it must be the identity. Since rotations leave γ invariant, this implies $R ( \unicode{x3bb} ) = \unicode{x3bb} $ , which is a contradiction. Therefore, λ has to be a circle.

Lemma 6.2. Let $G \subset {\mathrm {Diff}}^{1}_{}(\mathbb {S}^2)$ be a sharply 3-transitive homogeneous group of diffeomorphisms. Then, every $g \in G_2$ is conformal at the poles.

Proof. By Lemma 6.1, it suffices to consider the case of a planar diffeomorphism fixing the origin and mapping circles centered at the origin onto circles centered at the origin. We know that $A = \mathrm {D} g ( \mathbf {0} )$ is a linear isomorphism, so we may fix z _m and z _M respectively minimizing and maximizing $|Az |$ over 𝕊¹. Hypothesis then yields:

$$ \begin{align*}1 = \frac{ |g( t z_m ) | }{ |g( t z_M ) | } = \frac{ |A ( t z_m ) + o ( | t z_m | ) | }{ |A ( t z_M ) + o ( |t z_M | ) | } \rightarrow \frac{ | A ( z_m ) | }{ | A ( z_M ) | } \quad \text{as } t \rightarrow 0^{+} \text{.}\end{align*} $$

Therefore, $A ( \mathbb {S}^1 )$ is a circle. Since A preserves orientation, this is enough to infer that it is a conformal matrix. In other words, g is conformal at 0.

Before proceeding to the proof of Theorem C, let us make a small remark: if G is a 2-transitive homogeneous group of diffeomorphisms and $\delta> 0$ is given, we may obtain $h_{\delta } \in G$ such that $h_{\delta }$ fixes 0, but not ∞, and $\mathrm {D} h_{\delta } ( \mathbf {0} )$ is δ-close to id.

Indeed, by 2-transitivity, we may fix $h \in G$ such that $h ( \mathbf {0} ) = \mathbf {0}$ and $h ( \mathbf {\infty } ) = \mathbf {1}$ . We then let $h_{\delta } :=h^{-1} \circ R_t \circ h$ , for a sufficiently small t. This might seem like an underuse of the 3-transitivity hypothesis. However, as it turns out, a result from [9] implies any 2-transitive homogeneous group to actually be 3-transitive.

6.1 Proof of Theorem C

Assume—for the sake of contradiction—that G contains a non-conformal map g. By precomposing and postcomposing with suitable rotations, it may be assumed that g fixes 0 and that $A = \mathrm {D} g ( \mathbf {0} )$ is a non-conformal matrix. This means that the angle α between some pair $(u_0,v_0)$ of unit vectors is different from the angle β between their images. Given $\varepsilon = |\beta - \alpha |/2$ , fix $\delta> 0$ such that $|u - w | < \delta $ and $|v - z | < \delta $ on 𝕊¹ imply $|{\mathrm {ang}}( u,v ) - {\mathrm {ang}}( w,z )| < \varepsilon $ . By the prior discussion, we may fix $h_{\delta } \in G$ such that:

• • $h_{\delta } (\mathbf {0}) = \mathbf {0}$ ;

• • $h_{\delta }^{-1} ( \mathbf {\infty } ) :=p_{\delta } \ne \mathbf {\infty }$ ;

• • $\mathrm {D} h_{\delta } ( \mathbf {0} )$ is ${\delta }/{2}$ -close to id.

Notice that, by Lemma 6.2, g cannot fix ∞, or it would be conformal at the South Pole. Therefore, the 3-transitivity of G yields $f_{\delta } \in G_2$ such that $f_{\delta } ( g (\mathbf {\infty }) ) = p_{\delta }$ . From Lemma 6.2, $\mathrm {D} f_{\delta } ( \mathbf {0} )$ is conformal. Letting $\tilde {g} :=h_{\delta } \circ f_{\delta } \circ g$ , also $\mathrm {D} \tilde {g} ( \mathbf {0} )$ is conformal. By the chain rule, $\mathrm {D} \tilde {g} ( \mathbf {0} ) = \mathrm {D} h_{\delta } ( \mathbf {0} ) \mathrm {D} f_{\delta } ( \mathbf {0} ) A$ . However, the choices of ε and δ imply the right-hand side of this expression not to preserve the angle between u ₀ and v ₀, which is a contradiction. Thus, such non-conformal $g \in G$ cannot exist, and $G \subset {\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ . Equality then follows from the sharp 3-transitivity of ${\mathrm {M}\ddot {\mathrm {o}}\mathrm {b}}_{}(\mathbb {S}^2)$ .