A parameter ASIP for the quadratic family

Abstract

Consider the quadratic family $T_a(x) = a x (1 - x)$ for $x \in [0, 1]$ and mixing Collet–Eckmann (CE) parameters $a \in (2,4)$. For bounded $\varphi $, set $\tilde \varphi _{a} := \varphi - \int \varphi \, d\mu _a$, with $\mu _a$ the unique acim of $T_a$, and put $(\sigma _a (\varphi ))^2 := \int \tilde \varphi _{a}^2 \, d\mu _a + 2 \sum _{i>0} \int \tilde \varphi _{a} (\tilde \varphi _{a} \circ T^i_{a}) \, d\mu _a$. For any mixing Misiurewicz parameter $a_{*}$, we find a positive measure set $\Omega _{*}$ of mixing CE parameters, containing $a_{*}$ as a Lebesgue density point, such that for any Hölder $\varphi $ with $\sigma _{a_{*}}(\varphi )\ne 0$, there exists $\epsilon _\varphi>0$ such that, for normalized Lebesgue measure on $\Omega _{*}\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$, the functions $\xi _i(a)=\tilde \varphi _a(T_a^{i+1}(1/2))/\sigma _a (\varphi )$ satisfy an almost sure invariance principle (ASIP) for any error exponent $\gamma>2/5$. (In particular, the Birkhoff sums satisfy this ASIP.) Our argument goes along the lines of Schnellmann’s proof for piecewise expanding maps. We need to introduce a variant of Benedicks–Carleson parameter exclusion and to exploit fractional response and uniform exponential decay of correlations from Baladi et al [Whitney–Hölder continuity of the SRB measure for transversal families of smooth unimodal maps. Invent. Math. 201 (2015), 773–844].

1 Introduction

1.1 Background and motivation

Let $(\Omega _{*},m_{*},\mathcal {F}_{*})$ be a probability space. We say that a sequence of measurable functions $\xi _i \colon \Omega _{*} \to {\mathbb R}$ , $i\ge 1$ satisfies the almost sure invariance principle (ASIP) with error exponent $\gamma <1/2$ if there exist a probability space $(\Omega _W, m_W,\mathcal {F}_W)$ supporting a (centred) one-dimensional Brownian motion W and a sequence of measurable functions $\eta _i \colon \Omega _W\to {\mathbb R}$ , $i\ge 1$ , such that:

1. (i) the random variables $\{\xi _i\}_{i\ge 1}$ and $\{\eta _i\}_{i\ge 1}$ have the same distribution. (By definition of the distribution of discrete-time real-valued stochastic processes, this means that for any $n\ge 1$ and any $\{y_i\in {\mathbb R}\mid 1\le i \le n\}$ , the joint probability that $\xi _i\le y_i$ for all $1\le i \le n$ coincides with the joint probability that $\eta _i\le y_i$ for all $1\le i \le n$ .)

2. (ii) Almost surely, $ |W(n)-\sum _{i=1}^{n}\eta _i|=O(n^\gamma ) $ as $n\to \infty $ .

Since a Brownian motion at integer times coincides with a sum of independent identically distributed (i.i.d.) Gaussian variables, the above definition can also be formulated as an almost sure approximation, with error $o(n^{\gamma })$ , by a sum of i.i.d. Gaussian variables.

It is a classical result (see [PS]) that if the $\{\xi _i\}$ satisfies the ASIP, then it satisfies the law of the iterated logarithm (LIL), the central limit theorem (CLT) and the functional CLT: Letting $\sigma ^2>0$ be the variance of the Brownian motion W (the expectation is zero by assumption) and denoting Lebesgue measure by m, the LIL says that

$$ \begin{align*} \limsup_{n\to\infty}\frac1{\sqrt{2n\log\log n}}\sum_{i=1}^n \xi_i(a) =\sigma\quad\text{for } m_{*}\text{-almost every (a.e.) } a\in \Omega_{*}, \end{align*} $$

and the CLT (for the functional CLT, see [DLS, Lemma 5.1]) says that

$$ \begin{align*} \lim_{n\to\infty}m_{*} \bigg( \bigg\{\, a \in \Omega_{*} \mid \frac1{\sigma\sqrt{n}} \sum_{i=1}^n\xi_i(a)\le y \,\bigg\} \bigg) =\frac1{\sqrt{2\pi}}\int_{-\infty}^ye^{-s^2/2}\, ds \quad \text{for all } y \in {\mathbb R}. \end{align*} $$

We consider $I=[0,1]$ and the quadratic family

$$ \begin{align*} T_a(x) = a x (1 - x),\quad x \in I,\, a\in (2,4]. \end{align*} $$

Denote by $c=1/2$ the critical point of $T_a$ and set $ c_{j}(a)=T_a^{j}(c)$ for $j\ge 1$ .

If $\liminf _{n \to \infty } n^{-1} \log \partial _x(T_a^n) (T_a(c))> 0$ , we say that a is a Collet–Eckmann (CE) parameter. If a is CE, then $T_a$ admits a unique absolutely continuous invariant probability measure (acim) $\mu _a=h_a\,dm$ . Our goal is to find a positive Lebesgue measure set $\Omega _{*}$ of CE parameters with a Lebesgue density point $a_{*}\in \Omega _{*}$ such that for any Hölder continuous function $\varphi \colon I\to {\mathbb R}$ with $\sigma _{a_{*}}(\varphi )\ne 0$ (see equation (1.2)), there exists $\epsilon _\varphi>0$ such that the ASIP holds for $m_{*}$ the normalized Lebesgue measure on $\Omega _{*}\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$ and

$$ \begin{align*} \xi_j(a):=\varphi_a(c_{j+1}(a)), \quad j\ge 0, \quad a\in\Omega_{*}\cap[a_{*}-\epsilon_\varphi, a_{*}+\epsilon_\varphi], \end{align*} $$

where $\varphi _a$ is a suitable normalization of $\varphi $ (see equation (1.6)). We follow the approach of Schnellmann [Sch], who developed this program for transversal families of piecewise expanding maps $T_a$ , for which $\Omega _{*}$ can be taken to be an interval.

Our main motivation is to extend to the quadratic family the method developed by de Lima and Smania [DLS] in the setting of piecewise expanding maps, to study linear and fractional response. (This method requires a functional central limit theorem, see [DLS, Lemma 5.1].)

We say that $T_a$ is mixing if it is topologically mixing on

$$ \begin{align*} K(a):=[c_2(a), c_1(a)]. \end{align*} $$

It will be convenient below to restrict to mixing maps $T_a$ . Tiozzo recently showed [Ti, Corollary 3.15] (his result holds in fact for more general unimodal maps) that $T_a$ is (strongly) mixing for its unique measure of maximal entropy (MME) if its topological entropy is greater than $\log (2)/2$ . If a is a CE parameter with strongly mixing MME, then $T_a$ is topologically mixing on $K(a)$ since the measure of maximal entropy has full support there. (Indeed, since $T_a$ has no homtervals if $a\in CE$ , it is conjugated to its piecewise linear model $F_a$ by a homeomorphism which maps the MME of $F_a$ to the MME of $T_a$ , and the MME of $F_a$ is absolutely continuous with a positive density on $[F^2_a(c), F_a(c)]$ .) Since the topological entropy of $T_4$ is equal to $\log 2$ , and the topological entropy of $T_a$ is non-decreasing and continuous (in fact Hölder continuous [Gu]) in a, there exists $a_{\mathrm {mix}}<4$ such that for all $a \in (a_{\mathrm {mix}}, 4]\cap CE$ , the map $T_a$ is topologically mixing on $K(a)$ , and $\mu _a$ is strongly mixing, with support $K(a)$ .

Melbourne and Nicol [MN] showed the ASIP in the phase space $x\in K(a)$ , setting $\xi _i=T^i_a(x)$ for a fixed CE map $T_a$ , using an induced uniformly expanding system (then [PS, §7] provides an ASIP which projects to the ASIP for the original CE map). However, to the best of our knowledge, the ASIP in the parameter a is still open.

In the parameter space, typicality (the law of large numbers, LLN) and the LIL are known: Avila and Moreira [AM2] showed that for Lebesgue almost every CE map $T_a$ , the critical point is typical for its unique absolutely continuous invariant measure $\mu _a=h_a dm$ :

(1.1) $$ \begin{align} \lim_{n\to\infty} \frac1n \sum_{i=1}^n \varphi (c_i(a)) = \int_0^1 \varphi \, d\mu_a \quad\text{for all } \varphi\in C^0. \end{align} $$

(Benedicks and Carleson established typicality in [BC1] for the Cantor set of CE parameters considered there.) For Hölder continuous $\varphi \colon I\to {\mathbb R}$ and a topological mixing CE parameter a, define $\sigma _a( \varphi )\ge 0$ by

(1.2) $$ \begin{align} ( \sigma_a(\varphi))^2 &:= \int_0^1 \bigg( \varphi-\int\varphi \,d\mu_a \bigg)^2 \, d\mu_a \end{align} $$

(1.3) $$ \begin{align} &\quad + 2\sum_{i>0} \int_0^1 \bigg( \varphi - \int\varphi \, d\mu_a \bigg) \bigg( \varphi - \int \varphi \, d\mu_a \bigg) \circ T^i_a \, d\mu_a, \end{align} $$

where the sum in equation (1.3) is finite because topological mixing (that is, the fact that the map is non-renormalizable) implies [KN] exponential mixing for the acim and Hölder continuous observables.

In a work in progress, Gao and Shen [GS2] show that, for Lebesgue-a.e. a in the set of mixing CE parameters, for every Hölder observable $\varphi $ , either $\sigma _a(\varphi )=0$ or the critical point c of $T_a$ satisfies the LIL for $\varphi $ , that is,

$$ \begin{align*} \limsup_{n\to\infty} \frac1{\sqrt{2n\log\log n}}\sum_{i=1}^n \bigg( \varphi(T_a^i(c))-\int\varphi \, d\mu_a \bigg) = \sigma_a(\varphi). \end{align*} $$

1.2 Statement of the ASIP (Theorem 1.1)

To state our main result, we need more notation and definitions. For $j\ge 0$ and $a \in (a_{\mathrm {mix}},4]$ , set

$$ \begin{align*} x_j(a)=c_{j+1}(a)=T^{j+1}_a(c), \quad T_a^{\prime}(x)=\partial_xT_a(x), \quad x_j^{\prime}(a)=\partial_a x_j(a). \end{align*} $$

The family $T_a$ is transversal at ${a_{*}}$ if (see [Ts1]) there exists $C\ge 1$ such that

(1.4) $$ \begin{align} \frac{1}{C}\le \bigg| \frac{x_j^{\prime}(a_{*})}{(T_{a_{*}}^j)^{\prime} (c_1(a_{*}))} \bigg| \le C\quad\text{for all } j\ge1. \end{align} $$

By [Ts2, Theorem 3], all CE parameters are transversal. We refer to [Ts1, ( $N{V}_t$ )] for an equivalent condition expressed in terms of the postcritical orbit.

The map $T_a$ is $(H_a, \kappa _a)$ -polynomially recurrent, for $\kappa _a\ge 1$ and $H_a\ge 1$ , if

(1.5) $$ \begin{align} |x_{j-1} (a) - c| = |T^j_a(c)-c| \ge \frac {1}{j^{\kappa_a}} \quad \text{for all } j \ge H_a. \end{align} $$

If $\inf _{j\ge 1} |T_a^j(c)-c|>0$ , then a is called a Misiurewicz parameter. Misiurewicz parameters are CE and thus transversal. Avila and Moreira [AM1] showed that, for any $\kappa _0>1$ , the set of parameters a which are $(H_a, \kappa _0)$ -polynomially recurrent for some $H_a$ has full measure in the set of CE parameters. The set of Misiurewicz parameters a is uncountable (it has full Hausdorff dimension [Za, Theorem 1.4] but zero Lebesgue measure).

Finally, we introduce the normalization $\varphi _a$ : let $\varphi $ be bounded such that $\sigma _{a}(\varphi )\ne 0$ for a mixing CE parameter a. Then the function

(1.6) $$ \begin{align} \varphi_a(x):=\frac1{\sigma_a(\varphi)} \bigg(\varphi(x)-\int_0^1\varphi \, d\mu_a \bigg) \, \end{align} $$

is well defined and satisfies

(1.7) $$ \begin{align} \sigma_a(\varphi_a)=1\,\quad\text{and} \quad \int \varphi_a \, d\mu_a=0. \end{align} $$

Theorem 1.1. (Main Theorem: ASIP)

For any Misiurewicz parameter $a_{*}\in (a_{\mathrm {mix}},4)$ , there exists a positive Lebesgue measure set $\Omega _{*}$ of mixing polynomially recurrent parameters, containing $a_{*}$ as a Lebesgue density point, such that for any Hölder continuous function $\varphi $ with $\sigma _{a_{*}}(\varphi )\ne 0$ , there exists $\epsilon _\varphi>0$ such that the functions

(1.8) $$ \begin{align} \xi_n(a) :=\varphi_a (x_{n}(a))= \varphi_a (T_a^{n+1}(c)), \quad n \geq 1, \end{align} $$

satisfy the ASIP for normalized Lebesgue measure $m_{*}$ on $\Omega _{*} \cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$ and all error exponents $\gamma>2/5$ . (The choice of $\epsilon _\varphi $ ensures in particular that $\sigma _a(\varphi )\ne 0$ if $\sigma _{a_{*}}(\varphi )\ne 0$ .)

The value $a_{*}=4$ is not covered by our arguments for technical reasons, since $c_1$ and $c_2$ then lie on the boundary of I. (For example, the Banach space of [BBS] requires that the function on level zero of the tower be supported in $(0,1)$ , so this proof cannot cover the case $a=4$ .) It is possible (but a bit cumbersome) to handle (a one-sided neighbourhood of) this value by a change of coordinates as in [Ts1, Lemma 2.1].

We expect that the methods of this paper can be extended to the case when the ‘root’ $a_{*}$ is mixing, but only Collet–Eckmann and polynomially recurrent (for large enough $\kappa _0>1$ ), instead of Misiurewicz. (For example, [A, Lemma 8.1] would replace [DMS, Lemma V.6.5] in the proof of Proposition 2.2.) We restrict here to Misiurewicz parameters $a_{*}$ for the sake of simplicity. What is most desirable, in view of our original motivation to extend the analysis of [DLS], is to obtain a ‘fatter’ Cantor set $\Omega _{*}$ (as opposed to a fatter set of root points $a_{*}$ ). Indeed, this extension will probably require the ASIP on a set $\widetilde \Omega $ for which there exist $\beta>1$ and a full measure subset $\widetilde \Omega _1\subset \widetilde \Omega $ such that

(1.9) $$ \begin{align} \lim_{\epsilon \to 0} \frac{m([a-\epsilon, a+ \epsilon] \setminus \widetilde \Omega)}{\epsilon^\beta} = 0 \quad\text{for all } a \in \widetilde \Omega_1. \end{align} $$

(See [BS2, equation (5), Proposition F], note that [BS2, Lemma E] even uses $\beta <2$ close to $2$ .) Equation (1.9) is known for all $\beta <2$ for the sets $\widetilde \Omega _1\subset \widetilde \Omega $ studied by Tsujii [Ts1]. (Beware that Tsujii’s result cannot be used immediately. In particular, the main argument in the construction of the parameter set in Theorem 1 of ‘Pre-threshold fractional susceptibility function: holomorphy and response formula’, arXiv:2203.07942, is flawed.) For our Cantor set $\Omega _{*}\subset \Omega _\mathrm{BC}$ , we expect that for any $\kappa>1$ , taking $\kappa _0$ large enough in Proposition 2.2, the factor $\epsilon ^\beta $ in equation (1.9) must be replaced by $\epsilon |\log \epsilon |^{-\kappa }$ (see equation (2.20)), which does not seem good enough. Attaining the goal of our original motivation may thus require establishing the ASIP on a Cantor set having larger density, and thus weakening the polynomial lower recurrence in the construction (see comments in the next paragraph). We view this as the most desirable improvement of our main theorem.

To clarify the role of $\Omega _{*}$ , it is useful to compare Schnellmann’s proof with ours. In [Sch], Schnellmann studies suitable transversal one-parameter families of piecewise expanding interval maps and obtains a parameter ASIP on a set $\Omega _{*}$ which is just an interval $[0,\epsilon ^\varphi ]$ of parameters. Indeed, existence of an exponentially mixing acim enjoying fractional response (with uniform bounds) holds in an entire interval $[0,\epsilon ^\varphi ]$ in his setting [Sch, Proposition 4.3, Lemma 4.5]. So, $[0,\epsilon ^\varphi ]$ is the baseline parameter space for his analysis. Some parameters in this baseline cause difficulties (‘exceptionally small sets’), but Schnellmann can get away with just ignoring them (taking advantage of the fact that their total measure is controlled [Sch, part (III), Theorem 3.2, Lemma 4.1, proof of Lemmas 6.1 and 6.2]) instead of excluding them from the baseline. Our situation is different, since we need to exclude parameters which do not have an acim or for which exponential mixing or fractional response (with uniform bounds) does not hold. Our baseline set is a Cantor set, and the best we can do is to make it as fat as possible.

The polynomial recurrence in equation (1.5) in our parameter exclusion (Proposition 2.2), which causes the ‘thinness’ of $\Omega _{*}$ , is needed to apply the results of [BBS] in §§2.4 and 2.5 (Propositions 2.5 and 2.6 on uniform decorrelation and fractional response, and its consequence, Lemma 2.8). (See also (4.7) and (4.22), which may cause a different error exponent.) Due to this, we already exclude the parameters which could have exceptionally small image and we do not need to ignore them (Lemma 2.3, compare also [Sch, proof of Lemma 6.1] with equation (4.13) below). In addition, we get an easy proof of the local distortion estimate in equation (2.31). If the required consequences of [BBS] could be extended to sets of parameters which enjoy only exponential recurrence bounds, then we could use the (fatter) Benedicks–Carleson Cantor set $\Omega ^\varphi _\mathrm{BC}$ as a baseline instead of $\Omega _{*}$ (if necessary, the Benedicks–Carleson technique could be replaced by ideas from Tsujii [Ts1], Avila and Moreira [AM1] or Gao and Shen [GS1]). Next, one could try to ignore the parameters with exceptionally small images in Lemma 2.3. For equation (2.31), our proof is inspired from that of [DMS, Theorem V.6.2]. This is suboptimal but enough for our purposes. Adapting instead [DMS, Lemma V.6.4] could enhance equation (2.31).

We also note for the record here that the characteristic function $1_{\widetilde \Omega }$ of a fat enough Cantor set $\widetilde \Omega $ belongs to a Sobolev space $H^s_q(I)$ with $s>0$ (see [HM, Propositions 4.9 and 4.10]). Thus, working with a Cantor set of larger density may simplify some of our arguments (in the proof of Proposition 3.2, for example).

Finally, the results of this paper probably extend to more general families of smooth unimodal maps. In the present ‘proof of concept’ work, we choose to restrict to the quadratic family.

1.3 Structure of the text

Schnellmann pointed out [Sch, p. 370] that the ‘Markov partitions’ given by the intervals in the celebrated Benedicks–Carleson [BC1, BC2] parameter exclusion construction would be the key to extend his result to non-uniformly expanding interval maps.

Our paper carries out this plan and is organized as follows. After recalling basic facts in §2.1, we adapt in §2.2 the Benedicks–Carleson procedure to construct, in a neighbourhood of a topologically mixing Misiurewicz point $a_{*}$ , a sequence $\Omega _n\subset \Omega _{n-1}$ where $\Omega _n$ is a finite union of intervals in $\mathcal {P}_n$ . At each step, some intervals in $\mathcal {P}_n$ are partitioned and the intervals which do not satisfy a time-n polynomial recurrence assumption are excluded. The remaining Cantor set $\Omega _{*}(a_{*})=\bigcap _n \Omega _n$ is a positive Lebesgue measure set of parameters satisfying the Collet–Eckmann property, polynomial returns and distortion control, with uniform constants. (Our distortion bound in equation (2.31) is new.) In addition, the construction ensures that there are no ‘exceptionally small’ sets (Lemma 2.3). Applying results from [BBS], this ensures uniform exponential decay of correlations (Proposition 2.5) and fractional response (Proposition 2.6), from which we obtain regularity of the map $a\mapsto \sigma _a$ (Lemma 2.8).

Sections 3 and 4 contain the proof of the ASIP along the lines of [Sch]: first approximate the Birkhoff sum by a sum of blocks of polynomial size (§§4.1 and 4.2), then (§4.3) approximate these blocks by a martingale difference sequence $Y_j$ and apply Skorokhod’s representation theorem linking a martingale with a Brownian motion (see [PS, §3]). The usual application of the approach of [PS, Ch. 7] in dynamics uses a strong independence condition (see [PS, 7.1.2]) which we do not have (the $\xi _i$ terms are not iterations of a fixed map and there is no underlying invariant measure). (See the example (see [Ka, p. 646]) discussed in [Sch]. Also, as pointed out in [Sch], it is not clear how to apply the spectral techniques of [Go] in our setting.) We replace this strong independence condition by uniformity of constants in the exponential decay of correlations (given by [BBS]) which we translate into properties for the $\xi _i$ by switching from parameter to phase space (see Proposition 3.2), giving estimates similar to those in [PS, §3].

For $\varpi \in (0,1)$ , we shall denote by $C^\varpi $ the set of $\varpi $ -Hölder continuous functions $\varphi \colon I\to {\mathbb R}$ , putting $\|\varphi \|_\varpi =\sup |\varphi |+H_\varpi (\varphi )$ , with $H_\varpi (\varphi )$ the smallest $H_\varpi $ such that $|\varphi (x)-\varphi (y)|\le H_\varpi |x-y|^\varpi $ for all x, y in I. The letter C is used throughout to represent a (large) uniform constant, which may vary from place to place.

2 Bounds for the quadratic family. The Cantor set $\Omega _{*}(a_{*})$

2.1 Basic properties

Clearly, the maps

$$ \begin{align*} a\mapsto T^{\prime}_a(x)=\partial_xT_a(x)=a(1-2x),\quad x\mapsto\partial_a T_a(x)=x(1-x) \end{align*} $$

are Lipschitz continuous uniformly in $x\in I$ and $a\in (a_{\mathrm {mix}},4]$ , and, in addition,

(2.1) $$ \begin{align} \sup_{x\in I}|T_a^{\prime}(x)|\le \Lambda:=4 \quad \text{for all } a \in (a_{\mathrm{mix}},4]. \end{align} $$

Each $T_a$ has two monotonicity intervals, with partition points $0$ , $c=1/2$ and $1$ . The following easy lemma replaces [Sch], (30)]. (We do not need as in [Sch, (30)] that x has the same combinatorics under $T_{a_1}$ and $T_{a_2}$ up to the $(n-1)$ th iteration. We thus do not need any analogue of [Sch, Sublemma 5.4].)

Lemma 2.1. There exists $C<\infty $ such that, for any $a_1,a_2\in (2,4]$ , we have

(2.2) $$ \begin{align} |T_{a_1}^n(x)-T_{a_2}^n(x)|\le C\Lambda^{n}|a_1-a_2|\quad \text{for all } x\in I, \text{ for all } n \geq 1. \end{align} $$

Proof. Clearly, $|T_{a_1}(x)-T_{a_2}(x)|\le |a_1-a_2|$ . For $n\ge 2$ , using the definition in equation (2.1) of $\Lambda $ and setting $C=\sum _{j=0}^\infty \Lambda ^{-j}$ , we get

$$ \begin{align*} |T_{a_1}^n(x)-T_{a_2}^n(x)| &\leq |T_{a_1} (T_{a_1}^{n-1} (x)) - T_{a_2} (T_{a_1}^{n-1} (x))| +|T_{a_2} (T_{a_1}^{n-1} (x)) - T_{a_2} (T_{a_2}^{n-1} (x))| \\ &\leq |a_1-a_2| + \Lambda |T_{a_1}^{n-1} (x) - T_{a_2}^{n-1} (x)|\\ &\leq |a_1-a_2| (1 + \Lambda) + \Lambda^2 |T_{a_1}^{n-2} (x) - T_{a_2}^{n-2} (x)| \leq \cdots \\ & \leq |a_1-a_2| \sum_{j=0}^{n-1} \Lambda^j \leq C \Lambda^n |a_1-a_2|.\\[-41pt] \end{align*} $$

2.2 A polynomial Benedicks–Carleson construction ( $\Omega _{*}(a_{*})$ , $\mathcal {P}_n$ )

For each $j\ge 0$ , the function $x_j(a)=T_a^{j+1}(c)$ is a map from the parameter space $(a_{\mathrm {mix}},4]$ to the phase space $I=[0,1]$ , with $x_j(a)\in K(a)$ for all a. The transversality condition in equation (1.4) says that the derivatives of $x_j$ and $T_a^j$ are comparable at $a_{*}$ , so that statistical properties (such as the ASIP) can be transferred from the maps $x\mapsto T_a^j(x)$ to the maps $a\mapsto x_j(a)$ . To make this precise, we next construct a sequence of partitions in the parameter space. Our starting point is the following variant of the Benedicks–Carleson Cantor set $\Omega _\mathrm{BC}=\Omega _\mathrm{BC}(a_{*})$ (see [BC1, BC2]) associated to a Misiurewicz parameter $a_{*}$ (which is automatically transversal). (See equation (2.17) for the construction of $\Omega _\mathrm{BC}$ .)

Proposition 2.2. (The Cantor set $\Omega _{*}=\Omega _{*}(a_{*},\kappa _0)$ )

Let $a_{*}\in (a_{\mathrm {mix}},4]$ be a Misiurewicz parameter. There exist $\unicode{x3bb} _\mathrm{CE}\in (1, \Lambda )$ and $C_0\in (0,1)$ such that, for any $d_1\in (0,C_0\log \unicode{x3bb} _\mathrm{CE}/4)$ and $d_0>0$ , there exists $\epsilon>0$ such that for any $\kappa _0>1/d_1$ , for all large enough $N_0\ge 1$ , there exists a sequence $\mathcal {P}_j$ of finite sets of pairwise disjoint subintervals of

$$ \begin{align*} \omega_0 := [a_{*}-\epsilon, a_{*}+\epsilon] \cap (a_{\mathrm{mix}},4] \end{align*} $$

such that $\mathcal {P}_1=\mathcal {P}_2=\cdots =\mathcal {P}_{N_0}$ and, setting

$$ \begin{align*} \Omega_{*} = \Omega_{*}(a_{*},\kappa_0):=\bigcap_{j \geq N_0} \Omega_j \quad \text{with }\, \Omega_j:=\bigcup_{ \omega \in \mathcal{P}_j} \omega, \end{align*} $$

we have $\Omega _{j+1}\subset \Omega _j$ for $j\ge N_0$ , and (the first bound of equation (2.4) implies that $a\mapsto x_j(a)=T_a^{j+1}(c)$ is monotone on $\omega \in \mathcal {P}_j$ )

(2.3) $$ \begin{align} &\text{for all } j \geq 1, \text{ for all } \omega \in \mathcal{P}_j, \text{ for all } 0 \leq \ell < j , \text{ there exists } \omega^{\prime}\in \mathcal{P}_\ell \mbox{ such that } \omega \subset \omega^{\prime},\end{align} $$

(2.4) $$ \begin{align} & |x^{\prime}_j(a)|>0, |T_a^{j+1} (c) - c| > 0 \text{ for all } a \in \omega , \text{ for all } \omega \in \mathcal{P}_j, \text{ for all } j\ge 0, \end{align} $$

and there exists (note that equation (2.6) replaces [Sch, Lemma 2.4]) $C<\infty $ such that for all $j\geq N_0$ and $\omega \in \mathcal {P}_j$ ,

(2.5)

(2.6) $$ \begin{align} &\frac{1}{C}\leq \bigg|\frac{x_n^{\prime}(a)}{(T_a^{n})^{\prime}(T_a(c))} \bigg| \leq C\quad \text{for all } N_0 \leq n \leq j, \text{ for all } a \in \omega, \end{align} $$

(2.7)

and, moreover,

(2.8) $$ \begin{align} |T_a^{n+1}(c)-c|> n^{-\kappa_0}\quad \text{for all } N_0 \leq n \leq j, \text{ for all } a \in \omega. \end{align} $$

Finally, we have that $a_{*}\in \Omega _{*}$ is a Lebesgue density point of $\Omega _{*}$ , with

(2.9) $$ \begin{align} |\Omega_{*}|\ge (1-d_0 \cdot e_j)|\Omega_{j-1}| \quad \text{for all } j \geq N_0, \text{ where } e_j := \sum_{n = j}^\infty n^{-d_1 \cdot \kappa_0} , \end{align} $$

and we have the more precise (semi-local) bound

(2.10) $$ \begin{align} \sum_{\substack{\omega\in \mathcal{P}_{\ell}\\ \omega\subset \omega^{\prime}}} |\omega \setminus (\omega\cap \Omega_{*})| \le d_0 \cdot e_{\ell-\ell^{\prime}} | \omega^{\prime}| \quad \text{for all } \omega^{\prime} \in\mathcal{P}_{\ell^{\prime}}, \text{ for all } \ell \ge \ell^{\prime}\ge N_0. \end{align} $$

See Lemma 2.3 below regarding the absence of exceptionally small sets and §2.3 for a Hölder distortion property refining equation (2.16).

Clearly, equation (2.8) means that any $a\in \Omega _{*}$ is $(N_0,\kappa _0)$ -polynomially recurrent.

The bound in equation (2.9) implies that the Cantor set $\Omega _{*}$ has positive Lebesgue measure as soon as $d_1\cdot \kappa _0>1$ (and $N_0$ is large enough). Proposition 2.2 holds for such $\kappa _0$ , but we will need the stronger condition $d_1 \cdot \kappa _0 \geq 11/3$ to use equation (2.9) in the proof of Proposition 3.2 (and $d_1\cdot \kappa _0>9/5$ for Lemma 4.2).

The local bound in equation (2.10) is used in the proof of Lemma 4.1.

Proof of Proposition 2.2

Let $r_0 \ge 2$ be a large integer (to be chosen later, with $\epsilon \to 0$ as $r_0$ increases). For $r\ge r_0$ , set $I_r=I_r^{-}\cup I_r^{+}$ , where

$$ \begin{align*} I_r^{+} = [ c+e^{-r-1} ,c+ e^{-r} ),\quad I_r^{-} = ( c-e^{-r} , c-e^{-r-1} ],\quad U_r=(c-e^{-r},c + e^{-r}), \end{align*} $$

and cover each $I_r^{\pm }$ by $r^2$ pairwise disjoint intervals $I_{r,\ell }^{\pm }$ of equal size, each $I_{r,\ell }^{\pm }$ containing its boundary point closest to c. Let $\beta _\mathrm{BC}> \alpha _\mathrm{BC} > 0$ , where

$$ \begin{align*} e^{-n \alpha_\mathrm{BC}} \le n^{-\kappa_0} \quad \text{for all } n \ge N_{0}, \end{align*} $$

for $N_{0}$ a large integer to be chosen later. (The constant $\alpha _\mathrm{BC}$ is usually called $\alpha $ , but we shall need the letter $\alpha $ for another purpose in equation (2.30).)

For $a\in (a_{\mathrm {mix}},4]$ , $\nu \ge 1$ , and $r\ge r_0$ such that $T^\nu _a(c)\in I_r$ , the binding time $p(a)=p(r,a, \nu )$ of $U_r$ with $T^\nu _a(c)$ is the maximal $p\in {\mathbb Z}_+ \cup \{\infty \}$ such that

$$ \begin{align*} |T^{j}_a (x) - T^{j+\nu}_a (c)| \leq e^{-j\beta_\mathrm{BC} }\quad \text{for all } 1 \leq j\leq p, \text{ for all } x\in U_r. \end{align*} $$

The first free return time $\nu _1(a)$ of $a\in (a_{\mathrm {mix}},4]$ is the smallest integer $j \geq 1$ for which $T_a^j(c) \in U_{r_0}$ . For an interval $\omega \subset (a_{\mathrm {mix}},4]$ , the first free return time $\nu _1(\omega )$ is the smallest integer $j \geq 1$ for which there exists $a\in \omega $ with $T_a^j(c) \in U_{r_0}$ . If there exists $r=r(\omega )$ such that $x_{\nu _1-1}(\omega )\subset I_r$ (recall that $T^{\nu _1}_a(c)=x_{\nu _1-1}(a)$ ), we define the first binding time of $\omega $ by $p_1(\omega ) = \operatorname \min _{a \in \omega }p(r,a,\nu _1(\omega ))$ . For $i\ge 2$ , define inductively the ith free return time of (suitable) $\omega $ to be the largest integer $\nu _{i}(\omega )>\nu _{i-1}(\omega )+p_{i-1}(\omega )+1$ such that

$$ \begin{align*} T^{j}_a(c)\cap U_{r_0}=\emptyset \quad \text{for all } \nu_{i-1}(\omega)+p_{i-1}(\omega)+1\le j<\nu_{i}(\omega),\text{ for all } a \in \omega, \end{align*} $$

and, for $r(\omega )$ such that $x_{\nu _{i-1}-1}(\omega )\subset I_r$ , set the ith binding time of $\omega $ to be

$$ \begin{align*} p_{i}(\omega) = \operatorname\min_{a \in \omega} p(r,a,\nu_{i-1} (\omega)). \end{align*} $$

(Similarly, define inductively for $i\ge 2$ and a such that $ T^{\nu _{i-1}}_{a}(c)\in I_r$ , the pointwise binding times $p_i(a)$ and free returns $\nu _i( a)$ .) The iterates between $\nu _i(\omega )$ and $\nu _i(\omega )+p_i(\omega )$ form the ith bound period of $\omega $ , those between $\nu _{i-1}(\omega )+p_{i-1}(\omega )+1$ and $\nu _{i}(\omega )-1$ form its ith free period. Finally, if there exist $a\in \omega $ and $j\ge \nu _1(\omega )$ such that $T_a^j(c)\in U_{r_0}$ , we say that j is a return time of $\omega $ . (Return times either are free returns $\nu _i(\omega )$ or they occur during the bound period.)

Note that for any fixed $\epsilon $ , setting $\omega _0=[a_{*}-\epsilon ,a_{*}+\epsilon ]$ , there exists $N_\epsilon $ such that $x_{N_\epsilon }(\omega _0)$ contains a neighbourhood of c (indeed, by transversality, for any $a\in \omega _0\setminus \{a_{*}\}$ , there exists $N(a)$ such that $T_{a_{*}}^{N(a)+1}(c)$ and $T_{a}^{N(a)+1}(c)$ lie on different sides of c). In particular, $\nu _1(\omega _0)<\infty $ . Similarly, all $\nu _i(\omega _0)$ and $p_i(\omega _0)$ are finite.

Let $W_{a_{*}}$ be a neighbourhood of c disjoint from $\{T^n_{a_{*}}(c)\mid n\ge 1\}$ . From now on, we only consider $r_0$ large enough such that $\overline U_{r_0-1}\subset W_{a_{*}}$ . Set $W_{a_{*},r_0}^{+}=W_{a_{*}}\cap [c+e^{-r_0},1]$ and $W_{a_{*},r_0}^{-}=W_{a_{*}}\cap [0,c-e^{-r_0}]$ . We claim that for any fixed large $r_0$ , we have that $x_{\nu _1(\omega _0)-1}(\omega _0)$ contains $W_{a_{*},r_0}^{+}$ or $W_{a_{*},r_0}^{-}$ for all small enough $\epsilon $ . (This fact is used in [DMS, Lemma V.6.8]. There, $W_{a_{*}}$ is mistakenly mentioned instead of $W^{\pm }_{a_{*},r_0}$ . Our $r_0$ is denoted by $\Delta $ and our $x_n(a)$ is denoted $\xi _{n+1}(a)$ in [DMS].) Indeed, $x_{\nu _1(\omega _0)-1}(\omega _0)$ is an interval intersecting $U_{r_0}$ , and $x_{\nu _1(\omega _0)-1}(\omega _0)$ contains $T_{a_{*}}^{\nu _1(\omega )}(c)\notin W_{a_{*}}$ .

For small $\epsilon>0$ (to be chosen depending on $r_0$ ), the sequence $ \mathcal {P}_j$ can now be defined inductively: start with the single interval $ \mathcal {P}_0= \mathcal {P}_1=\cdots = \mathcal {P}_{N_0} =\{\omega _0 \}$ for $\epsilon $ small enough such that $\nu _1( \omega _0)\ge N_0$ (note that $\nu _1(\omega _0)$ increases if $r_0$ increases or $\epsilon $ decreases). (We refer throughout to [DMS, §V.6]. The original ideas and key estimates appeared previously in the work of Benedicks and Carleson [BC1, BC2]. The original construction in [BC1, BC2] is for $a_{*}=4$ , see [Mo] for a self-contained account. It extends to Misiurewicz parameters: for CE parameters, the condition in [DMS, Theorem 6.1] is equivalent to equation (1.4), taking large enough k in the last line of [DMS, p. 406, Step 2].)

For $j> N_0$ , each $ \omega \in \mathcal {P}_{j-1}$ is partitioned into finitely many (possibly just one) intervals, at least one of which will be included into an auxiliary partition $\mathcal {P}^{\prime }_{j}$ , as follows.

If j is not a free return time of $ \omega $ , we include $\omega $ in $\mathcal {P}^{\prime }_{j}$ . (That is, either j is not a return, or it is a return within the bound period.) If j is a free return time of $\omega $ but $x_{j-1}( \omega )$ does not contain an interval $I^{\pm }_{r,\ell }$ (we call this an inessential (free) return), we also include $\omega $ in $\mathcal {P}^{\prime }_{j}$ .

Otherwise, j is a free return time of $ \omega $ such that $x_{j-1}( \omega )$ contains at least one interval $I_{r,\ell }^{\pm }$ . We call this an essential (free) return. In that case, we decompose $x_{j-1}( \omega )$ into the following intervals:

$$ \begin{align*} x_{j-1}( \omega)\setminus U_{r_0},\quad \{x_{j-1}( \omega)\cap I_{r,\ell}^{\pm} \mid r \geq r_0,\, 1\leq \ell \leq r^2\}. \end{align*} $$

If $x_{j-1}( \omega )\setminus U_{r_0}\ne \emptyset $ , but any of the (at most two) connected components of $x_{j-1}( \omega )\setminus U_{r_0}$ has size less than $e^{-r_0}(1-1/e) r_0^{-2}=|I^{\pm }_{r_0,\ell }|$ , we join it to its neighbour $x_{j-1}(\omega )\cap I^{\pm }_{r_0,\ell }=I^{\pm }_{r_0,\ell }$ . If a connected component of $x_{j-1}( \omega )\setminus U_{r_0}$ has size larger than $S:=\sqrt {|U_{r_0}|}$ , we subdivide it into pairwise disjoint intervals of lengths between $S/2$ and S. If $x_{j-1}(\omega )\cap I_{r,\ell }^{\pm }\ne \emptyset $ , but $I_{r,\ell }^{\pm }$ is not contained in $x_{j-1}(\omega )$ (this can happen for at most two intervals $I_{r,\ell }^{\pm }$ ), we join $x_{j-1}(\omega )\cap I_{r,\ell }^{\pm }$ to its neighbour $x_{j-1}( \omega )\cap I^{\pm }_{r^{\prime },\ell ^{\prime }}=I^{\pm }_{r^{\prime },\ell ^{\prime }}$ . Denote by $\{\hat \omega _{r,\ell }\mid r\ge r_0-1\}$ the partition of $x_{j-1}(\omega )$ thus obtained, where the index $(r,\ell )$ refers to the ‘host’ interval $I_{r,\ell }$ contained in $\hat \omega _{r,\ell }$ if $r\ge r_0$ , while $\hat \omega _{r_0-1,\ell }\subset I\setminus U_{r_0}$ . Then we discard all intervals $\hat \omega _{r,\ell }$ for which

(2.11) $$ \begin{align} e^r\ge (j-1)^{\kappa_0}. \end{align} $$

Mapping the remaining intervals via the inverse of the diffeomorphism (see [DMS, Proposition V.6.2]) $x_{j-1}$ gives finitely many subintervals of $\omega $ which we include in $\mathcal {P}^{\prime }_j$ . Further intervals $\hat \omega _{r,\ell }$ need to be discarded from $\mathcal {P}^{\prime }_j$ , using a requirement denoted $(FA_{j})$ or $(FA^{\prime }_j)$ in [DMS, §V.6], [Mo], which finally defines $\mathcal {P}_j$ . For further use, we denote these remaining intervals by

(2.12) $$ \begin{align} \omega_{r,\ell}=x_{j-1}^{-1}(\hat\omega_{r,\ell}). \end{align} $$

It is well known (the original construction in [BC1, BC2] is for $a_{*}=4$ , see [Mo] for a self-contained account. It extends to Misiurewicz parameters: for CE parameters, the condition in [DMS, Theorem 6.1] is equivalent to equation (1.4), taking large enough k in the last line of [DMS, p. 406, Step 2]) [BC1, BC2, DMS] that, if we replace the condition in equation (2.11) (used to discard intervals) by the exponential condition

(2.13) $$ \begin{align} \omega \cap I_{r,\ell}^{\pm}\ne \emptyset \quad \mbox{and} \quad e^r\ge e^{\alpha_\mathrm{BC} (j-1)} , \end{align} $$

to construct sequences $\mathcal {P}_j^{\prime ,\mathrm{BC}}$ and $\mathcal {P}_j^\mathrm{BC}$ , then there exists $\unicode{x3bb} _\mathrm{CE}>1$ (called $e^\gamma $ in [DMS, (V.6.4), Theorem V.6.2]) such that for any small enough $\beta _\mathrm{BC}>\alpha _\mathrm{BC}>0$ , there exist $N_0^{\prime }$ such that if $r_0$ is large enough and $\epsilon>0$ small enough, then the $\mathcal {P}_j^\mathrm{BC}$ satisfy equations (2.3)–(2.7) (equation (2.5) is called $(EX_j)$ in [DMS, §V.6]) for some $C<\infty $ , and the following condition (noted $(BA_j)$ in the literature) holds for all $j\ge N_0^{\prime }$ :

(2.14) $$ \begin{align} 2 |T_a^{n+1}(c)-c|> e^{-n \alpha_\mathrm{BC}} \quad\text{for all } N^{\prime}_0 \leq n \leq j, \text{ for all } a \in \omega \text{ for all } \omega \in \mathcal{P}_j^{\prime,\mathrm{BC}}. \end{align} $$

(Strictly speaking, the condition $(BA_j)$ does not involve the factor $2$ , and a condition $(BA^{\prime }_j)$ requiring that for each $\omega \in \mathcal {P}_j^{\prime \mathrm{BC}}$ , there exists $a\in \omega $ with $|T_a^{n+1}(c)-c|> e^{-n \alpha _\mathrm{BC}}$ for $N_0^{\prime }\le n\le j$ is used in some lemmas. See [DMS, §V.6, Step 5].) Since $\unicode{x3bb} _\mathrm{CE}$ does not depend on $\alpha _\mathrm{BC}$ , $N_0$ or $N_0^{\prime }$ , we may assume that

and we may replace $N_0$ by $\operatorname \max \{N_0, N_0^{\prime }\}$ .

In particular [DMS, Proposition V.6.1, Lemma V.6.1(b), (c)] give $\gamma _0>0$ , $\unicode{x3bb} _\mathrm{CE}=e^\gamma \in (1, e^{\gamma _0})$ and $C_0>0$ (independent of $r_0$ and $\epsilon $ ) such that if $a\in \Omega _n$ and $\nu _{\ell +1}(a)\le n$ , writing $p_\ell $ , $\nu _\ell $ for $p_\ell (a)$ , $\nu _\ell (a)$ , we have

(2.15)

To establish equation (2.5) (the bound below will also be used for equation (2.34)), one takes $r_0$ such that

The key distortion bound [DMS, Proposition V.6.3] gives C such that

(2.16) $$ \begin{align} \bigg| \frac{x_j^{\prime}(a_1)}{x_j^{\prime}(a_2)} \bigg| \leq C \quad \text{for all } N_0\le j \le n , \text{ for all } a_1, a_2\in\omega , \end{align} $$

whenever $n+1$ is a free return time of $\omega \in \mathcal {P}_n$ with $x_{n+1}(\omega )\subset U_{r_0/2}$ . The bound in equation (2.6) follows from [DMS, Proposition V.6.2 and Theorem V.6.2].

Let $\Omega ^{\prime }_j:=\bigcup _{ \omega \in \mathcal {P}^{\prime }_j} \omega $ , recall $\Omega _j$ , and define $\Omega _j^\mathrm{BC}$ and $\Omega _j^{\prime ,\mathrm{BC}}$ accordingly, setting

(2.17) $$ \begin{align} \Omega_\mathrm{BC}=\Omega_\mathrm{BC}(a_{*},\alpha_\mathrm{BC})=\bigcap_j \Omega^\mathrm{BC}_j \quad \mbox{so that } \Omega_{*}(a_{*})\subset\Omega_\mathrm{BC}(a_{*}). \end{align} $$

It is easy to check that equation (2.11) implies equation (2.8) (for returns during a bound period, use that $\ell ^{-\kappa _0}-e^{-\ell \beta _\mathrm{BC }}\ge j^{-\kappa _0}$ for all $N_0\le \ell \le j-1$ , up to increasing $N_0$ again). Our choice of $N_0$ implies $\Omega _j\subset \Omega _j^\mathrm{BC}$ . Also, equation (2.6) with equation (2.4) imply that all points in $\Omega _{*}$ are transversal. Since equation (2.7) is an immediate consequence of equations (2.5) and (2.6), it only remains to establish that $a_{*}$ is a Lebesgue density point in $\Omega _{*}$ (clearly, $a_{*}\in \Omega _{*}$ ) and that equations (2.9) and (2.10) hold.

To show that $a_{*}$ is a Lebesgue density point of $\Omega _{*}$ , we may follow [DMS, Step 7 of the proof of Theorem V.6.1], replacing $C e^{-i C_0}$ there by $C^{\prime } i^{-\kappa _0}$ . (We mention a typo there: although the constant $C=C(\epsilon )$ in the unnumbered equation on [DMS, p. 433] tends to zero as $\epsilon =|\omega _0|/2\to 0$ , the constant $C_0$ is (fortunately) uniformly bounded away from zero. See the proof of [DMS, Lemma V.6.5].)

We next establish equations (2.9) and (2.10). For suitably small $\bar \eta>0$ , and for $J_0\ge 1$ such that $\prod _{j=J_0}^\infty (1-e^{-\bar \eta j})>3/4$ (since $\bar \eta $ is independent of $\epsilon $ , $r_0$ , $N_0$ , we may take $N_0\ge J_0$ ), the parameter exclusion rule in equation (2.13) gives $d^{\prime }_0>0$ (tending to zero with $\epsilon $ ) such that ([DMS, §V.6, Step 7], [Mo, §6])

(2.18) $$ \begin{align} \begin{cases} |\omega \cap \Omega^{\prime,\mathrm{BC}}_j|\ge (1- d^{\prime}_0 e^{-j \bar \eta})|\omega| &\text{for all } \omega \in \mathcal{P}^\mathrm{BC}_{j-1}, \text{ for all } j\ge J_0,\\ |\Omega^\mathrm{BC}_j|\ge |\Omega^{\prime,\mathrm{BC}}_j|-e^{-j\bar \eta}|\omega_0| & \text{for all } j\ge J_0. \end{cases} \end{align} $$

The above implies $|\Omega ^\mathrm{BC}_j|\ge (1- d^{\prime }_0 e^{-\bar \eta j})|\Omega ^\mathrm{BC}_{j-1}|-e^{-\bar \eta j}|\omega _0|$ for $j\ge J_0$ , and, exploiting that $|\omega _0|=|\Omega ^\mathrm{BC}_n|$ for all $n\le N_0$ with $N_0\ge J_0$ , and using the definition of $J_0$ , also that

$$ \begin{align*} |\Omega^\mathrm{BC}_j| \geq \bigg( \prod_{n=J_0}^j (1- d^{\prime}_0 e^{-\bar \eta n}) - \sum_{n=J_0}^j e^{-\tilde \eta n} \bigg ) |\omega_0| \geq \frac 1 2 |\omega_0| \quad \text{for all } j \geq J_0. \end{align*} $$

(By taking larger $J_0$ , that is, smaller $\epsilon $ , we could replace $1/2$ by a number close to $1$ .) Thus, applying inductively

$$ \begin{align*} |\Omega^\mathrm{BC}_j|\geq ( (1- d^{\prime}_0 e^{-\bar \eta j})- 2 e^{-\bar \eta j} ) |\Omega^\mathrm{BC}_{j-1}| \quad \text{for all } j\geq J_0 , \end{align*} $$

we find $\tilde \eta>0$ such that for any $j\ge J_0$ ,

(2.19) $$ \begin{align} |\Omega_\mathrm{BC}|\geq \prod_{n=j}^\infty (1 - ( d^{\prime}_0+2) e^{-\bar \eta n}) |\Omega_{j-1}^\mathrm{BC}| \geq (1 - (\tilde d_0 + 2) e^{-\tilde \eta j} ) |\Omega_{j-1}^\mathrm{BC}|. \end{align} $$

Recall that we fixed $d_1 \in (0, {C_0}/{4} \log \unicode{x3bb} _\mathrm{CE})$ (independently of $\kappa _0$ ). Let $J_1$ be such that $\prod _{j=J_1}^\infty (1-e^{-\bar \eta j}-j^{-2})>3/4$ and return to the sets $\Omega _j$ , $\Omega _j^{\prime }$ constructed using the (polynomial) exclusion rule in equation (2.11) for $\kappa _0>1/d_1$ . We claim that for any $d_0>0$ , if $\epsilon $ is small enough,

(2.20) $$ \begin{align} \begin{cases} |\omega \cap \Omega^{\prime}_j|\ge (1-d_0 \cdot j^{-d_1 \kappa_0}) |\omega| &\text{for all } \omega \in \mathcal{P}_{j-1}, \text{ for all } j\ge J_1,\\ | \Omega_j|\ge |\Omega^{\prime}_j|- e^{-j\bar \eta}|\omega_0| &\text{for all } j\ge J_1. \end{cases} \end{align} $$

Before establishing this claim, we note that, mutatis mutandis, equation (2.20) combined with the arguments leading to equation (2.19) implies equation (2.9), while the more precise claim in equation (2.10) follows from the refinement of equation (2.20) coming from the second statement of [DMS, Lemma V.6.9] (see the use of [Mo, Lemma 6.3] in [Mo, Lemma 6.4 and Proposition 6.5]).

To show equation (2.20), we proceed in three steps, performing the necessary changes in the proof in [DMS, §V.6]. Recall equation (2.12).

First, up to taking larger $N_0$ , the conclusion of [DMS, Lemma V.6.5] (which deals with $(BA^{\prime }_{j})$ for $\omega \in \mathcal {P}_{j-1}$ satisfying $(BA^{\prime }_{j-1})$ and $(EX_{j-1})$ and having a return at time j), if we replace the exponential rate $(BA^{\prime }_{j-1})$ there by our polynomial rate in equation (2.8), becomes

(2.21) $$ \begin{align} \frac{|\omega \setminus \bigcup_{r \geq \kappa_0\log j}\, \omega_{r,\ell}|}{|\omega|}\ge 1- C j ^{-d_1 \kappa_0}\quad \text{for all } j \geq N_0. \end{align} $$

To show this first claim, use that the constant $C_0 \in (0,1)$ (introduced above) is independent of $\kappa _0$ (because $\unicode{x3bb} _\mathrm{CE}$ does not depend on $\kappa _0$ ) and that [DMS, Lemma V.6.1] gives that the bound period p of a free return $\nu <j$ with

(2.22) $$ \begin{align} I_{r^{\prime},\ell^{\prime}}\subset x_\nu(\omega) \quad \mbox{for } r_0\leq r^{\prime} \leq \kappa_0\log \nu\leq \kappa_0 \log j, \end{align} $$

satisfies $p\ge C_0 r^{\prime }$ . Then, up to taking larger $N_0$ , we can replace [DMS, V.(6.20)] in the proof of [DMS, Lemma V.6.5] by

(2.23)

where we used $d_1 \leq ({C_0}/{4}) \log \unicode{x3bb} _\mathrm{CE}$ in the second inequality. We can thus replace the chain of inequalities after [DMS, V.(6.20)] (using the distortion bound in equation (2.16) for $\tilde \omega \subset \omega $ the largest interval with $x_n(\tilde \omega )\subset U_{r_0/2}$ , taking $\epsilon $ small enough and $N_0$ large enough such that equation (2.23) also holds for $\tilde \omega $ ) by

$$ \begin{align*} \frac{|\bigcup_{r \ge \kappa_0\log j}\, \omega_{r,\ell}|}{|\omega|} \le \frac{|\bigcup_{r \ge \kappa_0\log j}\, \omega_{r,\ell}|}{|\tilde \omega|} \le C\frac 1{j^{\kappa_0}}\frac{1}{|x_j(\tilde \omega)|} \le C j^{-d_1 \cdot \kappa_0}. \end{align*} $$

Second, [DMS, Lemma V.6.6] (which deals with $(FA_j)$ ) uses equation (2.14) only via [DMS, Lemma V.6.3], while [DMS, Lemma V.6.3] still holds (with the same proof) if we replace equation (2.14) by our stronger assumption in equation (2.8). (We mention here a typo: [DMS, V.(6.24)] follows from [DMS, V.(6.22)] (and not [DMS, V.(6.20)] as stated there).)

Third, [DMS, Lemmas V.6.7–6.9] are unchanged, establishing equation (2.20).

Lemma 2.3 is the analogue of [Sch, (III)′]).

Lemma 2.3. (No exceptionally small sets)

For any $\kappa _1>\kappa _0$ , there exists $N_1\ge N_0$ such that $|x_j(\omega )|> j^{-\kappa _1}$ for all $j\ge N_1$ and $\omega \in \mathcal {P}_j=\mathcal {P}_j(a_{*},\kappa _0)$ .

Proof. We first show the lemma assuming that there exists $d_2\in (0,1)$ such that for any $j\ge N_0$ and any $ \omega \in \mathcal {P}_{j}$ , we have

(2.24) $$ \begin{align} |x_j(\omega)| \geq \frac{d_2 e^{-r_0}(1-1/e)}{( \kappa_0\log j)^2 j^{\kappa_0}} , \end{align} $$

with $r_0$ as in the proof of Proposition 2.2. Indeed, equation (2.24) implies that

$$ \begin{align*} |x_{j}(\omega)|\geq \frac{d_2 e^{-r_0}(1 - e^{-1})}{\kappa_0^2} \frac{1}{j^{\kappa_0} (\log j)^2} \quad \text{for all } \omega\in\mathcal{P}_j, \text{ for all } j\ge N_0. \end{align*} $$

Clearly, there exists $N_1(\kappa _1)\ge N_0$ such that the right-hand side is larger than $j^{-\kappa _1}$ for all $j\ge N_1$ .

To establish equation (2.24), we shall use equation (2.8). If $j+1$ is an essential free return time of $\omega $ , then taking r minimal such that $x_{j}(\omega )$ contains an interval $I_{r,\ell }^{\pm }$ ,

(2.25) $$ \begin{align} |x_{j}(\omega)|\ge |I_{r,\ell}^{\pm}|=e^{-r}\frac {1-1/e}{r^2}> \frac{j^{-\kappa_0}(1-1/e)}{( \kappa_0\log j)^2}. \end{align} $$

Otherwise, letting $j^{\prime }+1=\nu _{i^{\prime }}(\omega )\ge \nu _1(\omega )$ be the largest essential free return time of $\omega $ such that $ j^{\prime }+1<j+1$ , we have $\omega \in \mathcal {P}_{j^{\prime }}$ (since if $\tilde {\omega } \supset \omega $ , $\tilde {\omega } \in \mathcal {P}_{j^{\prime }}$ , then $\tilde {\omega }$ is never cut between time $j^{\prime }$ and j), so that equation (2.25) implies

$$ \begin{align*} |x_{j^{\prime}}( \omega)|> \frac{1-1/e}{( \kappa_0\log j^{\prime})^2(j^{\prime})^{\kappa_0}}>\frac{1-1/e}{( \kappa_0\log j)^2j^{\kappa_0}}. \end{align*} $$

We shall combine the above bound with [DMS, Lemma V.6.3 and Propositions V.6.1 and V.6.2] to handle the three cases left, namely: the time $j+1$ is an inessential free return of $\omega $ ; the time $j+1$ is a return within a bound period of $\omega $ ; and the intersection of $x_j(\omega )$ and $U_{r_0}$ is empty.

If $j+1=\nu _i( \omega )$ is an inessential free return, then [DMS, V.(6.15) in Lemma V.6.3] gives, for $i^{\prime }\le i$ as defined above,

(2.26) $$ \begin{align} |x_{j}(\omega)|\ge 2^{i-i^{\prime}} |x_{j^{\prime}}(\omega)|> 2^{i-i^{\prime}} \frac{1-1/e}{( \kappa_0\log j)^2j^{\kappa_0}}. \end{align} $$

If $j+1$ is a return within the bound period of a previous free return $j^{\prime \prime }+1$ of $\omega $ , then using equation (2.25) for the bound period of an essential return, respectively equation (2.26) for the bound period of a non-essential return, and applying the first claim of [DMS, Lemma V.6.3], we find $d_2\in (0,1)$ such that

(2.27)

If $x_j(\omega )\cap U_{r_0}=\emptyset $ , then [DMS, V.(6.2) in Propositions V.6.1 and V.6.2] and equation (2.25) give

(2.28) $$ \begin{align} |x_{j}( \omega)|\ge d_2 e^{-r_0} |x_{j^{\prime}}( \omega)|> \frac{d_2 e^{-r_0}(1-1/e)}{( \kappa_0\log j)^2j^{\kappa_0}}. \end{align} $$

We have shown equation (2.24) and thus Lemma 2.3.

2.3 A Hölder local distortion estimate

From now on, let $a_{*}\in (a_{\mathrm {mix}}, 4)$ be a Misiurewicz parameter, fix $\kappa _0\ge 11 / 3 d_1$ and let $\Omega _{*}=\Omega _{*}(a_{*},\kappa _0)\subset \Omega _\mathrm{BC}= \Omega _\mathrm{BC}(a_{*})$ be the positive measure Cantor set constructed in §2.2 via families $\mathcal {P}_j=\mathcal {P}_j(a_{*},\kappa _0)$ . The following replaces [Sch, equations (33) and (31)]. (For [Sch, equation (30)], see equation (2.2). We do not need [Sch, equation (32)].) The bound in equation (2.31) is new.

Lemma 2.4. (Hölder distortion bounds)

There exists $C<\infty $ such that for all $n\ge N_0$ (with $N_0$ as in Proposition 2.2) and any $\omega \in \mathcal {P}_n=\mathcal {P}_n(a_{*},\kappa _0)$ ,

(2.29) $$ \begin{align} \frac 1 C \leq \bigg|\frac{x_n^{\prime}(a)/x_j^{\prime}(a)}{(T_a^{n-j})^{\prime}(x_j(a))} \bigg| \leq C \quad \text{for all } 1 \leq j \leq n , \text{ for all } a \in \omega. \end{align} $$

In addition, there exist $C<\infty $ and $M_0>\kappa _0$ such that, for all $n\ge N_0$ , each $\tilde \omega \in \mathcal {P}_n=\mathcal {P}_n(a_{*},\kappa _0)$ , and every $\omega \subset \tilde \omega $ and $\alpha \in [0,1)$ satisfying

(2.30) $$ \begin{align} |x_{n}(\omega)| \leq n^{-M_0/(1-\alpha)} , \end{align} $$

we have

(2.31) $$ \begin{align} \bigg| \frac{x_n^{\prime}(a_1)}{x_n^{\prime}(a_2)} \bigg| \leq 1 + C|x_{n}([a_1,a_2])|^\alpha \quad \text{for all } a_1, a_2\in\omega. \end{align} $$

If $\alpha =0$ , and $n+1$ is a free return of $\omega \in \mathcal {P}_n$ , the bound in equation (2.31) is just equation (2.16). We shall require equation (2.31) for some $\alpha>0$ in Corollary 3.4.

Proof. The bound in equation (2.29) is an immediate consequence of equation (2.6).

We first claim that there exist $C^{\prime }$ and $\kappa _2>0$ such that for any n,

(2.32) $$ \begin{align} \sum_{i=0}^{j-1} |x_i(\omega)|\le C^{\prime} j^{\kappa_0+1+\kappa_2 } |x_{j}(\omega)| \quad \text{for all } 1\le j\le n ,\,\text{ for all } \omega\subset \tilde \omega \in \mathcal{P}_n. \end{align} $$

(Our proof is inspired from that of [DMS, Theorem V.6.2]. This is suboptimal but enough for our purposes. Adapting instead [DMS, Lemma V.6.4] could enhance equation (2.31).) To start, there is C such that for any $0\le i\le j\le n$ , using equations (2.3) and (2.29), there exists $a=a(i,j,\omega )\in \omega $ such that, setting $X_{i,j}= x_j \circ x_i^{-1}$ ,

(2.33) $$ \begin{align} \frac{|x_i(\omega)|}{ |x_{j}(\omega)| } &= \frac{|x_i(\omega)|}{ |X_{i,j} (x_{i}(\omega))|}= \frac{|x^{\prime}_i(a)|}{ |x^{\prime}_{j}(a)|} \le \frac{C}{ |(T^{j-i}_a)^{\prime}(T^{i+1}_a(c))| }. \end{align} $$

(We used $X^{\prime }_{i,j}=x^{\prime }_j/x^{\prime }_i$ and the mean value theorem in the second equality.) Next, let $s_j(a)$ be the largest $\ell $ with $\nu _{\ell }(a)\le j$ , and put

$$ \begin{align*} &q_\ell(a)=\nu_{\ell+1}(a)-(\nu_\ell(a)+p_\ell(a)+1), \quad \ell=0, \ldots, s_j(a)-1, \\[-2pt] &q_{s_j(a)}(a)=\operatorname\max\{0, j-(\nu_{s_j(a)}(a)+p_{s_j(a)}(a)+1)\} , \quad F_j(a)= \sum_{\ell=0}^{s_j(a)} q_\ell(a). \end{align*} $$

(The condition $(FA)_n$ implicitly used in Proposition 2.2 says that for some fixed arbitrarily small $\tau>0$ , $ F_\ell (a)\ge \ell (1-\tau )$ for $N_0\le \ell \le n$ . We shall not need this here.) Set $p_\ell =p_\ell (a)$ , $\nu _\ell =\nu _\ell (a)$ , $q_\ell =q_\ell (a)$ , and $s_j=s_j(a)$ . Assume first that $i=0$ . Then, we have (see e.g. [DMS, V.(6.11)])

$$ \begin{align*} \nonumber|(T^{j-i}_a)^{\prime}(T^{i+1}_a(c))|&=|(T^{j}_a)^{\prime}(T_a(c))|\\ \nonumber&=|(T^{\nu_{1}-1}_a)^{\prime}(T_a(c))| \cdot |(T^{j+1-\nu_{s_j}}_a)^{\prime}(T^{\nu_s}_a(c))|\\ \nonumber&\quad \cdot \bigg (\prod_{\ell=1}^{s_j-1}|(T^{p_{\ell}+1}_a)^{\prime}(T^{\nu_\ell}_a(c))| |(T^{q_\ell}_a)^{\prime}(T^{\nu_\ell+p_\ell+1}_a(c))| \bigg ). \end{align*} $$

Since a satisfies $(BA)_m$ and $(FA)_m$ for all $m\le n$ , the bounds in equation (2.15) give $\unicode{x3bb} _\mathrm{CE}$ , $\gamma _0>0$ and $C_0>0$ such that

Similarly, $|(T^{\nu _{1}-1}_a)^{\prime }(T_a(c))|>C_0 e^{\gamma _0 \nu _1}$ . Next, if $j \le \nu _{s_j}+p_{s_j}+1$ , we have

where we used equation (2.8) and [DMS, Lemma V.6.1.b, Proposition V.6.1]. If $j> \nu _{s_j}+p_{s_j}+1$ , we have, using [DMS, Lemma V.6.1.c, Proposition V.6.1],

Summarizing,

Since $p_\ell {\kern-1.2pt}\ge{\kern-1.2pt} C_0 r_0$ (see after equation (2.22)), we have $j{\kern-1.2pt}-{\kern-1.2pt}F_j{\kern-1.2pt}\ge{\kern-1.2pt} j C_0 r_0$ while $s_j{\kern-1.2pt}\le{\kern-1.2pt} j/(C_0r_0)$ . We took $r_0$ large enough (see after equation (2.15)) such that

(2.34)

Finally, using the trivial bound $e^{\gamma _0 F_{j}(a)}\ge 1$ , we find

$$ \begin{align*} |(T^{j}_a)^{\prime}(T_a(c))|\ge \frac{ j^{-\kappa_0}}{C}. \end{align*} $$

If $i\ge 1$ and $\nu _{\ell _i}(a)+p_{\ell _i}(a)<i<\nu _{\ell _i+1}(a)$ for some $\ell _i\ge 1$ , then we proceed as for $i=0$ , replacing $|(T^{\nu _{1}-1}_a)^{\prime }(T_a(c))|$ by $|(T^{\nu _{\ell _i+1}-i}_a)^{\prime }(T^{i+1}_a(c))|$ and setting $F_{i,j}(a)=\nu _{\ell _i+1}(a)-i+\sum _{\ell \ge \ell _i+1}^{s_j(a)} q_\ell (a)$ . Then,

We have $j-i-F_{i,j}\ge (j-i) C_0 r_0$ , while $s_j-s_i\le (j-i)/(C_0r_0)$ , and we find, using $e^{\gamma _0 F_{i,j}(a)}\ge 1$ (we do not know or need $F_{i,j}(a)\ge (1-\tau )(j-i)$ ),

$$ \begin{align*} |(T^{j-i}_a)^{\prime}(T^{i+1}_a(c))|\ge\frac{ j^{-\kappa_0}}{C}. \end{align*} $$

Otherwise, $\nu _{\ell _i}(a)\le i-1\le \nu _{\ell _i}(a)+p_{\ell _i}(a)$ for some $\ell _i\ge 1$ . There may be (non-free) returns during the $\ell _i$ th bound period. To bypass this difficulty, we exploit that the length of the $\ell $ th bound period is of the order r if $x_{\nu _{\ell }}(a)\in I_{r}$ [DMS, Lemma V.6.1a]. By equation (2.11), we have $r_{\ell _i}=O(\log (\nu _{\ell _i}))\le C \kappa _0 \log i$ . Thus, the missing factor in the $\ell $ th bound period is $\le \Lambda ^{ C\kappa _0\log i}\le i^{\kappa _2}$ , and

$$ \begin{align*} |(T^{j-i}_a)^{\prime}(T^{i+1}_a(c))|\ge \frac{ j^{-\kappa_0}}{ C i ^{\kappa_2 }}. \end{align*} $$

Summing over i, and recalling equation (2.33), this establishes equation (2.32).

Next, taking $a_1,a_2 \in \omega $ , note that equation (2.7) (using the first bound of equation (2.4) if $i<N_0$ ) implies that for all $N_0\le i\le n$ , recalling $T^{\prime }_a(x)=a(1-2x)$ ,

(2.35) $$ \begin{align} \nonumber |T_{a_1}^{\prime}(x_i(a_1))&-T_{a_2}^{\prime}(x_i(a_2))|\\ \nonumber & \leq |T_{a_1}^{\prime}(x_i(a_1)) - T_{a_2}^{\prime} (x_i (a_1))| + |T_{a_2}^{\prime} (x_i (a_1)) - T_{a_2}^{\prime} (x_i (a_2))| \\ &\leq |a_1-a_2| + 2 a_2 |x_i(a_1) - x_i(a_2)| \leq (C + 2 a_2) |x_i(\omega)|. \end{align} $$

(Note that equation (2.35) replaces [Sch, (36)].) We claim that there exists $C^{\prime \prime }$ with

(2.36) $$ \begin{align} \bigg|\frac {(T_{a_1}^j)^{\prime}(x_0(a_1))}{(T_{a_2}^j)^{\prime}(x_0(a_2))}\bigg| \leq 1 + C^{\prime\prime} e^{C^{\prime\prime}} j^{2\kappa_0+1+\kappa_2} |x_{j}(\omega)| \, \quad\text{for all } 1\leq j\leq n. \end{align} $$

(The above replaces [Sch, (37)].) Indeed, using the classical bound

$$ \begin{align*} \prod_{i=0}^{j-1} (1+ \upsilon_i) \leq \exp \bigg( \sum_{i=0}^{j-1}\upsilon_i \bigg) \leq 1+ e^{\sum_{i=0}^{j-1}\upsilon_i}\sum_{i=0}^{j-1}\upsilon_i\quad \mbox{if all }\upsilon_i\geq 0, \end{align*} $$

we have, setting $C^{\prime \prime }=2C^{\prime }C(C+2a_2 )$ ,

(2.37) $$ \begin{align} \nonumber \bigg| \frac{(T_{a_1}^j)^{\prime} (x_0 (a_1))}{(T_{a_2}^j)^{\prime} (x_0 (a_2))} & \bigg| = \bigg| \prod_{i=0}^{j-1} \frac{T_{a_1}^{\prime} (x_i (a_1))}{T_{a_2}^{\prime} (x_i (a_2))} \bigg| \\ \nonumber &\leq 1 + e^{\sum_i |-1 + {T_{a_1}^{\prime} (x_i (a_1))}/{T_{a_2}^{\prime} (x_i (a_2))} |} \cdot \sum_{i=0}^{j-1} \bigg|\frac{T_{a_1}^{\prime} (x_i (a_1))}{T_{a_2}^{\prime} (x_i (a_2))} - 1 \bigg| \\ \nonumber &\leq 1 + e^{(C + 2 a_2) \sum_{i=0}^{j-1} C |x_i(\omega)| i^{\kappa_0}} \cdot (C + 2 a_2) \sum_{i=0}^{j-1} C |x_i(\omega)| i^{\kappa_0} \\ &\leq 1 + e^{C^{\prime\prime} j^{2\kappa_0+1+\kappa_2} |x_{j}(\omega)|} \cdot C^{\prime\prime} j^{2\kappa_0+1+\kappa_2} |x_{j}(\omega)| \quad\text{for all } j\leq n, \end{align} $$

where we used equations (2.35) and (2.8) (the first bound of equation (2.4) if $i<N_0$ ) in the second inequality, and equation (2.32) in the last inequality. Setting $M_0:=4\kappa _0+3+2\kappa _2$ , if equation (2.30) holds for $\omega $ , then equation (2.32) gives for all $N_0\le j \leq n$ ,

$$ \begin{align*} C^{\prime\prime} j^{2\kappa_0 +1+\kappa_2} |x_{j}(\omega)| & \leq C^{\prime\prime} j^{2\kappa_0+1+\kappa_2} C^{\prime} j^{\kappa_0+1+\kappa_2}|x_{n}(\omega)|\\ &\leq C^{\prime} \frac{j^{3\kappa_0+3+2\kappa_2}}{n^{M_0/(1-\alpha)}} \le C^{\prime\prime\prime}. \end{align*} $$

This proves equation (2.36). Similarly, $ |{ (T_{a_2}^j)^{\prime }(x_0(a_2))}/{(T_{a_1}^j)^{\prime }(x_0(a_1))}| \leq 1+C^{\prime \prime } e^{C^{\prime \prime }} j^{2\kappa _0+1+\kappa _2} |x_{j}(\omega )|$ . Therefore,

$$ \begin{align*} \bigg |\frac1{(T_{a_1}^j)^{\prime}(x_0(a_1))}-\frac1{(T_{a_2}^j)^{\prime}(x_0(a_2))}\bigg | \leq C^{\prime\prime\prime}\frac{j^{2\kappa_0+1+\kappa_2} |x_{j}(\omega)|}{|(T_{a_1}^j)^{\prime}(x_0(a_1))|}. \end{align*} $$

We can then adapt the end of the proof of [Sch, (31)]. Comparing each term on the right-hand side of

$$ \begin{align*} \frac{x_n^{\prime}(a)}{(T_a^n)^{\prime}(x_0(a))} =x_0^{\prime}(a)+\sum_{j=1}^n\frac{(\partial_aT_a)(x_{j-1}(a))}{(T_a^j)^{\prime}(x_0(a))}\quad \text{for all } a\in\tilde \omega\in \mathcal{P}_n, \end{align*} $$

for $a=a_1$ and $a=a_2$ , we find, since $x_0^{\prime }(a)=\partial _ac_1(a)=1/4$ and

$$ \begin{align*} |\partial_aT_a|_{a_1}(x_{j-1}(a_1))-\partial_aT_a|_{a_2}(x_{j-1}(a_2))| \leq |x_{j-1}(a_1)-x_{j-1}(a_2)|\leq |x_{j-1}(\omega)|, \end{align*} $$

recalling equation (2.5), and applying equation (2.32) and then equation (2.30) for $M_0=4\kappa _0+3+2\kappa _2$ ,

Finally, we have, using equation (2.6) (which plays the role of [Sch, Lemma 2.4]),

$$ \begin{align*} \bigg| \frac{x_n^{\prime}(a_1)}{x_n^{\prime}(a_2)} \bigg| & \leq \bigg| \frac{(T_{a_1}^n)^{\prime} (x_0(a_1))}{(T_{a_2}^n)^{\prime} (x_0 (a_2))} \bigg| \bigg( 1 + \tilde C |x_{n} (\omega)|^\alpha \frac{|(T_{a_2}^n)^{\prime}(x_0(a_2))|}{|x_n^{\prime}(a_2)|} \bigg) \\ & \leq 1 + C \tilde C |x_{n} (\omega)|^\alpha.\\[-34pt] \end{align*} $$

2.4 Uniform decorrelation and Hölder response

The maps $x_j$ are not the iterates of a fixed dynamical system admitting an invariant measure. To exploit statistical information on the iterates of the mixing CE map $(T_{a_0},\mu _{a_0})$ , we will ‘switch locally’ from $x_j$ to $T_{a_0}^j$ (see Lemma 3.3), using that any $a\in \Omega _{*}$ satisfies the following uniform decorrelation result for Hölder continuous observables. (The factor $\|\varphi \|_{L_1(dm)}$ in the right-hand side of [Sch, Proposition 4.3] is replaced in Proposition 2.5 by $\|\varphi \|_{L^1(d\mu _a)}\le \|\varphi \|_{L^\infty (dm)}$ . This does not impact the use of [Sch, p. 36, Proposition 4.3].) For $q>1$ and $s\in [0, 1/q)$ , we denote by $H^s_q(I)=F^s_{q,2}(I)$ the Sobolev space of functions of differentiability s and integrability q supported in I (see [RS]).

Proposition 2.5. (Uniform decay of correlations)

For any $s>0$ and $q>1$ , there exist $C<\infty $ and $\rho ^s_q<1$ such that for all $\varphi \in H^s_q(I)$ , $\psi \in L^\infty (dm)$ , $a\in \Omega _{*}(a_{*}, \kappa _0)$ ,

$$ \begin{align*} \bigg| \!\int_0^1 \varphi (\psi\circ T_a^n) \, dm - \!\int_0^1\varphi \, dm \!\int_0^1\psi \, d\mu_a \bigg| \leq C \|\varphi\|_{H^s_q}\|\psi\|_{L^1(d\mu_a)}(\rho^s_q)^n\quad\text{for all } n\geq 1. \end{align*} $$

For any $\varpi>0$ , there exist $C<\infty $ and $\rho _\varpi <1$ such that for all $\varphi \in C^\varpi $ , $\psi \in L^\infty (dm)$ , $a\in \Omega _{*}(a_{*}, \kappa _0)$ ,

$$ \begin{align*} \bigg| \!\int_0^1 \!\varphi (\psi\circ T_a^n)\, d\mu_{a } - \!\int_0^1 \!\varphi \, d\mu_{a} \!\int_0^1 \!\psi\, d\mu_{a} \bigg| \le C \|\varphi\|_\varpi \|\psi\|_{L^1(d\mu_a)} (\rho_\varpi)^{n}\quad \text{for all } n\ge 1. \end{align*} $$

We also use Hölder bounds on $a\mapsto \mu _a$ as a distribution (in Lemma 2.8).

Proposition 2.6. (Fractional response)

For any $\Theta \in (0,1/2)$ , there exists C such that for all $\varphi \in C^{1/2}$ ,

(2.38) $$ \begin{align} \bigg| \!\int \varphi \, d\mu_a - \!\int \varphi \, d\mu_{a^{\prime}} \bigg| \leq C |a-a^{\prime}|^\Theta \|\varphi\|_{1/2} \quad \text{for all } a, a^{\prime}\in \Omega_{*} (a_{*},\kappa_0). \end{align} $$

Our proof of Proposition 2.5 uses the following facts.

Sublemma 2.7. For any $a\in \Omega _\mathrm{BC}$ , the density $h_a$ of $\mu _a$ lies in $H^s_q(I)$ for all $s\in [0,1/2)$ and $q \in (1, 2/(1+2s))$ . In addition, for any $(H_0,\kappa _0)$ polynomially recurrent $a_{*}$ , there exists $C_{s,q,a_{*}}<\infty $ such that

$$ \begin{align*} \sup_{a\in \Omega_{*}{a_{*},\kappa_0}} \|h_a\|_{H^s_q(I)}\le C_{s,q,a_{*}}. \end{align*} $$

Proof. In the Misiurewicz case, the first claim is [Se, Theorem 10], using Ruelle’s [Ru, Theorem 9, Remark 16.a] decomposition of $h_a$ into the sum of a $C^1$ function and an exponentially decaying sum of ‘spikes’ $x\mapsto |x-c_k(a)|^{-1/2}$ and square root singularities $x\mapsto |x-c_k(a)|^{1/2}$ . For a general $a \in \Omega _\mathrm{BC}$ , set $ T^{-k}_{a,\varsigma }:=(T_a^k|_{U_{k,a,\varsigma }})^{-1}$ for $k\ge 1$ and $\varsigma \in \pm $ , where $U_{k,a,\varsigma }$ is the monotonicity interval of $T_a^k$ containing c, located to the right of c for $\varsigma =+$ , to the left of c for $\varsigma =-$ . Then, since we assumed $\unicode{x3bb} _\mathrm{CE}>e^{ 14 \alpha _\mathrm{BC}}$ in the proof of Proposition 2.2, use [BS1, Proposition 2.7] that there exist a $C^1$ function $\psi _a \colon I \to {\mathbb R}_+$ and $C^\infty $ functions $\Xi ^k_{a,\pm } \colon [0,1] \to [0,1]$ supported in a neighbourhood of $c_k(a)$ in $T^k_a(U_{k,a,\pm })$ , such that

(2.39) $$ \begin{align} h_a(x)=\psi_a(x) + \sum_{k = 1}^{\infty}\sum_{\varsigma\in \{+,-\}}\chi_{k,a}(x) \frac{\Xi^k_{a,\varsigma}(T^{-k}_{a,\varsigma}(x)) \psi_a(T^{-k}_{a,\varsigma}(x)) }{|(T_a^{k})^{\prime}(T^{-k}_{a,\varsigma}(x))|} , \end{align} $$

where $\chi _{k,a}(x)=1_{\pm x<\pm c_k(a)}$ if $\pm T_a^k$ has a local maximum at c. Setting $\Psi :=\Xi ^k_{a,\varsigma } \cdot \psi _a$ , we find $C^1$ functions $\Psi _{k,\ell }$ for $\ell =1,2,3$ , with

(2.40) $$ \begin{align} \frac{\Psi (T^{-k}_{a,\varsigma}(x)) }{|(T_a^{k})^{\prime}(T^{-k}_{a,\varsigma}(x))|} = \frac{\Psi_{k,1}(x)}{|x-c_k(a)|^{1/2} } + \Psi_{k,2}(x) |x-c_k(a)|^{1/2} + \Psi_{k,3}(x) \end{align} $$

for any $x\in \operatorname {supp} (\chi _{k,a})$ . Finally, use [Se, Lemmas 11 and 12].

For the second claim, it is convenient to use an alternative decomposition of $h_a$ . First, recall that [BBS, Corollary 1.6] gives a set $\Omega _{\mathrm {slow}}$ of full measure in the set of mixing CE parameters such that for any $\tilde {a} \in \Omega _{\mathrm {slow}}$ and each $\kappa _0>1$ , there exist $H_0\ge 1$ and a set $\Delta _0(\tilde {a}, \kappa _0)\subset \Omega _{\mathrm {slow}}$ of $(H_0,\kappa _0)$ -polynomially recurrent (and thus transversal) parameters, with $\tilde {a}$ as a Lebesgue density point, such that Proposition 2.5 holds for all $a \in \Delta _0$ . (It is unknown whether $\tilde a\in \Delta _0$ .) The proof involves constructing a tower for each parameter in $\Delta _0$ . We claim that, up to reducing the value of $\epsilon $ in the proof of Proposition 2.2, we can replace $\tilde a$ by $a_{*}$ and $\Delta _0$ by $\Omega _{*}(a_{*}, \kappa _0)$ . Indeed, $\Delta _0$ was constructed in [BBS, Proposition 2.1], and it suffices to observe that the required uniformity in constants is satisfied by equations (2.5) and (2.8), while [BBS, equations (8) and (7)] are exactly [DMS, V.(6.1), V.(6.2) in Proposition V.6.1].

Let then

$$ \begin{align*} \Pi_a (\hat \psi)(x)=\sum_{j \ge 0, \varsigma \in\pm} \frac{ \unicode{x3bb}^j} {|(T_a^j)^{\prime}(T^{-j}_{a,\varsigma}(x))|} \psi_j(T^{-j}_{a,\varsigma}(x)) \end{align*} $$

(for a suitable $\unicode{x3bb}>1$ ) be the projection from the tower with polynomial recurrence used in [BBS], and let $\hat {\mathcal {L}}_{a}$ be the lift $\mathcal {L}_a \Pi _a=\Pi _a \hat {\mathcal {L}}_a$ of the transfer operator $\mathcal {L}_a \varphi (x)=\sum _{T_a(y)=x} \varphi (y)/|T^{\prime }_a(y)|$ . Then ([BBS, (66)] gives uniform Lasota–Yorke estimates, [BBS, Lemmas 3.8, 4.5 and 4.6, Proposition 4.1] give the weak norm bounds needed by Keller and Liverani [KL]), there exist $C<\infty $ and $\theta <1$ such that, letting $\|\cdot \|^{\prime }_a$ be the norm of the Sobolev space $\mathcal {B}_a^{W^1_1}$ of [BBS],

(2.41) $$ \begin{align} \|\hat {\mathcal{L}}_a^n(\hat \varphi)-\hat h_a \hat \nu(\hat \varphi)\|^{\prime}_a\le C \|\hat \varphi\|^{\prime}_a \theta^n \quad \text{for all } \hat \varphi \in \mathcal{B}_a^{W^1_1}, \text{ for all } a\in \Omega_{\tilde{a}}, \end{align} $$

where $\hat h_a$ is the fixed point (the fixed point property determines $\hat h_a$ by its value on the level zero of the tower) of $\hat {\mathcal {L}}_a$ on $\mathcal {B}_a^{W^1_1}$ normalized by $\int \Pi _a \hat h_a \, dm = 1$ , while $\hat \nu $ , the non-negative measure whose density with respect to Lebesgue in the level j of the tower is $\unicode{x3bb} ^j$ , is the fixed point of the dual of $\hat {\mathcal {L}}_a$ (see [BS1, equation (85)], note that $\nu (\hat h_a)=1$ is automatic). Since $\Pi _a\hat h_a=h_a$ and the $W^1_1$ norm dominates any $H^s_q$ norm on I if $s\in [0,1)$ and ${1< q<1/s}$ (by the Sobolev embedding, more precisely [RS, Ch. 2], the bounded inclusions $ W^1_1 \subset W^\sigma _1=F^{\sigma }_{1,1}\subset F^{\sigma }_{1,2}\subset F^{s}_{q,2}=H^s_q$ , if $\sigma =1+s-1/q\in (0,1)$ and $q\in (1,\infty )$ ), the decomposition in equation (2.40) combined with the uniform bound in equation (2.41) (for $\hat \varphi $ vanishing on all levels $\geq 1$ and constant on level zero of the tower, with $\hat \nu (\varphi )=1$ ) gives the second claim of the sublemma, using again [Se, Lemmas 11 and 12].

Proof of Proposition 2.5

Recall from the proof of Proposition 2.2 that we have ${\unicode{x3bb} _\mathrm{CE}>e^{14 \alpha _\mathrm{BC}}}$ . By mollification, it is enough to prove both bounds for $C^1$ functions $\varphi $ . It is in fact enough to show the first bound for $\varphi \in C^1$ as follows. Indeed, again by mollification (see e.g. the proof of [Se, Lemma 14]), if the first bound holds for $\varphi \in C^1$ , then it holds for any $\varphi \in H^s_q(I)$ with $q>1$ and $s>0$ . Therefore, since the density $h_a$ of $\mu _a$ lies in $H^s_q(I)$ for all $s\in (0,1/2)$ and $q \in (1, 2/(1+2s))$ by Sublemma 2.7 (with norm uniformly bounded in a), the second bound follows from the first bound for $\varphi \in C^1$ (using that $C^1$ functions are bounded multipliers on $H^s_q$ ).

Next, we observed in the proof of Sublemma 2.7 that we can replace the set called $\Delta _0$ in [BBS, Corollary 1.6] by $\Omega _{*}(a_{*}, \kappa _0)$ . The first bound for Lipschitz continuous $\varphi $ thus follows from the second assertion of [BBS, Corollary 1.6], since $\Omega _{*}\subset [a_{\mathrm {mix}}, 4)$ . Indeed, note first that a is topologically mixing if and only if its renormalization period $P_{a}$ is equal to one. Second, observe that the constant $C_{\varphi , \psi }$ in the second claim of [BBS, Corollary 1.6] can be replaced by $C \|\varphi \|_\varpi \|\psi \|_{L^1(d\mu _a)}$ for a constant C uniform in a in view of [BBS, Lemmas 4.5 and 4.6] and the principle of uniform boundedness. More precisely, using the notation from the proof of Sublemma 2.7, we have

$$ \begin{align*} &\int (\psi \circ T_a^n) \varphi \, dm = \int \psi \Pi_a (\hat {\mathcal{L}}_a^n(\hat \varphi)) \, dm \quad \mbox{if } \Pi_a(\hat \varphi)=\varphi , \\ & \int (\psi \circ T_a^n) \varphi h_a \, dm= \int \psi \Pi_a(\hat {\mathcal{L}}_a^n(\hat \varphi_a)) \, dm \quad\mbox{if } \Pi_a(\hat \varphi_a)=\varphi h_a. \end{align*} $$

Since $\Pi _a\hat h_a=h_a$ , any Lipschitz continuous $\varphi $ can be written as $\Pi _a (\hat \varphi )$ (take $\hat \varphi _0= \varphi $ on the level zero and $\hat \varphi _j\equiv 0$ on levels $j\ge 1$ ) such that, on the one hand, $\|\hat \varphi \|_a^{\prime }\le C\|\varphi \|_1$ uniformly in a, and, on the other hand, $\hat \nu (\hat \varphi ) = \int \varphi \, dm$ . We conclude by applying equation (2.41) from the proof of Sublemma 2.7. (The Banach space of [BBS] requires that the function on level zero of the tower be supported in $(0,1)$ , so this proof cannot cover the case $a=4$ .)

Proof of Proposition 2.6

If $a=a_{*}$ , the bound is an immediate consequence of the first claim of [BBS, Corollary 1.6], since we can replace the set denoted $\Delta _0$ there by $\Omega _{*}(a_{*}, \kappa _0)$ , as observed in the proof of Sublemma 2.7 and used in the proof of Proposition 2.5. If $a\ne a_{*}$ , the uniformity of the constants given by Proposition 2.2 ensures that we may construct the reference tower in [BBS] at a (instead of $a_{*}$ ), viewing $a^{\prime }$ as a perturbation of a.

2.5 Hölder regularity of the variance $\sigma _a(\varphi )$

Propositions 2.5 and 2.6 will imply the following regularity of $a\mapsto \sigma _a(\varphi )$ on $\Omega _{*}$ .

Lemma 2.8. (Regularity of $\sigma _a(\varphi )$ )

For any $\varpi \in (0,1]$ , there exist $\theta \in (0,\operatorname \min \{1/2, \varpi \})$ and $C<\infty $ such that for each $\varphi \in C^\varpi $ with $\sigma _{a_{*}}(\varphi )>0$ , there exists $\epsilon _\varphi>0$ such that

$$ \begin{align*} C_{\epsilon_\varphi}(\varphi):=\inf_{a\in \Omega_{a_{*}}\cap [a_{*}-\epsilon_\varphi, a_{*}+\epsilon_\varphi]} \sigma_{a}(\varphi)>0, \end{align*} $$

and such that for all $a,a^{\prime }\in \Omega _{*}(a_{*},\kappa _0)\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$ , we have

(2.42) $$ \begin{align} |\sigma_a(\varphi)-\sigma_{a^{\prime}}(\varphi)|\le\frac{ C}{2 C_\epsilon(\varphi)} \|\varphi\|_\varpi |a-a^{\prime}|^\theta. \end{align} $$

Proof. Let $k_0>1$ be a large integer to be chosen at the end of the proof. By the second claim of Proposition 2.5, there exist $\rho =\rho _\varpi <1$ and $C_0$ such that

$$ \begin{align*} &\sum_{k> k_0} \bigg| \int \bigg( \varphi-\int\varphi \, d\mu_{a} \bigg) \cdot \bigg( \bigg( \varphi - \int\varphi \, d\mu_{a} \bigg) \circ T_{ a}^k \bigg) \, d\mu_{a} \bigg| \\ &\quad\le C_0 \|\varphi\|_\varpi^2 \cdot \frac{\rho^{k_0}}{1-\rho} \quad \text{for all } k_0\ge 1, \text{ for all } a \in \Omega_{a_{*}} , \text{ for all } \varphi \in C^\varpi. \end{align*} $$

Set $A_a=\int \varphi \, d\mu _a$ . Since $\int ((\varphi -A_a)\circ T_a^k )(\varphi -A_a) \, d\mu _a = \int (\varphi \circ T_a^k) \varphi \, d\mu _a -A_a^2$ , we have

$$ \begin{align*} |\sigma_a(\varphi)^2-\sigma_{a^{\prime}}(\varphi)^2| &\leq 2 \sum_{k=0}^{k_0-1} \bigg |\int\varphi(\varphi\circ T_a^k) \,d\mu_a - \int\varphi(\varphi\circ T_{a^{\prime}}^k) \, d\mu_{a} \bigg | \\ &\quad + 2\sum_{k=0}^{k_0-1} \bigg |\int\varphi(\varphi\circ T_{a^{\prime}}^k) \, d\mu_a - \int\varphi(\varphi\circ T_{a^{\prime}}^k) \, d\mu_{a^{\prime}} \bigg | \\ &\quad + 2\sum_{k=0}^{k_0-1}\bigg | \bigg( \int \varphi \, d\mu_a \bigg)^2 - \bigg( \int \varphi \, d\mu_{a^{\prime}} \bigg)^2 \bigg | \\ &\quad + 4 C_0 \|\varphi\|^2_\varpi\frac{\rho^{k_0}}{1-\rho} \quad \text{for all } k_0\ge 1. \end{align*} $$

Assume for a moment that $\varpi \ge 1/2$ . The $\varpi $ -Hölder constant of $\varphi (\varphi \circ T_{\bar a}^k)$ (for $\bar a=a$ or $a^{\prime }$ ) is bounded by $\Lambda ^{ k}\|\varphi \|^2_\varpi $ . Thus, Proposition 2.6 gives for any $\Theta <1/2$ a constant $C_1=C_1(\Theta )$ such that for $a, a^{\prime } \in \Omega _{a_{*}}$ and $\varphi \in C^\varpi $ ,

$$ \begin{align*} &|\sigma_a(\varphi)^2-\sigma_{a^{\prime}}(\varphi)^2| \leq k_0 C_1 \|\varphi\|^2_\varpi \Lambda^{ k_0}|a-a^{\prime}|^\Theta + C_0 \|\varphi\|^2_\varpi \frac{\rho^{k_0}}{1-\rho}\\ &\quad+ 2 \sum_{k=0}^{k_0-1} \bigg |\int\varphi(\varphi\circ T_a^k) \, d\mu_a - \int\varphi(\varphi\circ T_{a^{\prime}}^k) \, d\mu_{a} \bigg |\quad \text{for all } k_0\geq 1. \end{align*} $$

Next, equation (2.2) gives that

$$ \begin{align*} \int &|\varphi\circ T_a^k-\varphi\circ T_{a^{\prime}}^k |\, d \mu_a \leq \|\varphi\|_{\varpi} (C \Lambda^{k}|a-a^{\prime}|)^{\varpi}. \end{align*} $$

Therefore, we find

(2.43) $$ \begin{align} |\sigma_a(\varphi)^2-\sigma_{a^{\prime}}(\varphi)^2 | &\leq k_0 C_1 \|\varphi\|^2_\varpi \Lambda^{ k_0}|a-a^{\prime}|^\Theta + 4 C_0 \|\varphi\|^2_\varpi \frac{\rho^{k_0}}{1-\rho}\nonumber\\ &\quad+ k_0 \|\varphi\|_{\varpi} ( C\Lambda^{k_0}|a-a^{\prime}|)^{\varpi}. \end{align} $$

We conclude the proof for $\varpi \ge 1/2$ by dividing equation (2.43) by $|a-a^{\prime }|^{\theta }$ for small enough $\theta>0$ and optimizing in $k_0$ , using also $(\sigma _a - \sigma _{a^{\prime }}) (\sigma _a + \sigma _{a^{\prime }}) = \sigma _a^2 - \sigma _{a^{\prime }}^2$ .

If $\varpi \in (0,1/2)$ , mollification gives $\varphi _\upsilon \in C^{1/2}$ and $C_4$ such that

$$ \begin{align*} \|\varphi_\upsilon\|_{1/2} \leq C_4 \upsilon^{\varpi-1/2}\|\varphi\|_\varpi , \quad \sup |(\varphi\circ T_{\bar a}^k)\varphi - (\varphi_\upsilon \circ T_{\bar a}^k)\varphi_\upsilon| \leq C_4 \upsilon^{\varpi}\Lambda^k\|\varphi\|_\varpi \end{align*} $$

for all small $\upsilon>0$ , all $0\le k \le k_0$ and all $\bar a \in \Omega _{a_{*}}$ . To conclude, optimize in ${\upsilon =|a-a^{\prime }|^{\theta _0}}$ for small $\theta _0>0$ , taking $\theta $ smaller (in particular $\theta <\varpi \theta _0$ ).

3 Switching locally from the parameter to the phase space

Let $a_{*}$ , $\mathcal {P}_j(a_{*},\kappa _0)$ and $\Omega _{*}=\Omega _{*}(a_{*},\kappa _0)$ be as in Proposition 2.2 for $\kappa _0 \ge 11 / (3d_1)$ , and fix $\varpi \in (0,1)$ . This section is devoted to Proposition 3.2, the main estimate (analogous to [Sch, Proposition 5.1]) towards a law of large numbers for the squares of the blocks which will be defined in §4 (see Lemma 4.2).

From now on, fix $\varpi \in (0,1)$ and a $\varpi $ -Hölder continuous function $\varphi \colon I\to {\mathbb R}$ , recalling $\varphi _a$ , $\sigma _a(\varphi )$ from equations (1.6) and (1.3), and assume $\sigma _{a_{*}}(\varphi )>0$ . Lemma 2.8 gives $\epsilon _\varphi>0$ such that

(3.1) $$ \begin{align} \sigma_a(\varphi)>0 \quad \text{for all } a\in\Omega_{*}^\varphi:=\Omega_{*}\cap [a_{*}-\epsilon_\varphi, a_{*}+\epsilon_\varphi]. \end{align} $$

If $\epsilon _\varphi <\epsilon $ , we replace $\Omega _{*}$ by $\Omega _{*}^\varphi $ by replacing $\epsilon $ in the proof of Proposition 2.2 with $\epsilon _\varphi $ . (This is harmless as it can only improve the constants.)

Remark 3.1. ( $\theta $ -Hölder–Whitney extensions of $\varphi _a$ and $\xi _n(a)$ )

By Proposition 2.6, the function $a \mapsto \int \varphi \, d \mu _a$ is $\Theta $ -Hölder continuous on $\Omega _{*}$ for any $\Theta <1/2$ . By Lemma 2.8, the function $a \mapsto \sigma _a(\varphi )\ge 0$ is $\theta $ -Hölder continuous on $\Omega _{*}$ for some $\theta <\operatorname \min \{1/2, \varpi \}$ , and uniformly bounded away from zero on $\Omega _{*}^\varphi $ . Taking $\Theta \ge \theta $ , the map $a \mapsto \varphi _a(u) = (\varphi (u)-\int \varphi d\mu _a)/\sigma _a$ is $\theta $ -Hölder continuous on $\Omega _{*}^\varphi $ uniformly in $u\in I$ . By the Whitney extension theorem, we extend each map $a \mapsto \varphi _a(u)$ to a $\theta $ -Hölder continuous map on $[a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$ , uniformly in $u\in I$ . In addition, there exists $\widetilde C < \infty $ such that

(3.2) $$ \begin{align} \|\varphi_a\|_\infty\le \|\varphi_a\|_{\varpi} \le \widetilde C\|\varphi\|_{\varpi} \quad \text{for all } a\in [a_{*}-\epsilon_\varphi, a_{*}+\epsilon_\varphi]. \end{align} $$

Then, using equation (2.2), we may extend each map $a \mapsto \xi _n(a)=\varphi _a(T_a^{n+1}(c))$ to a $\theta $ -Hölder continuous map on $[a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$ , with $\theta $ -Hölder constant bounded by $C\Lambda ^{\theta (n+1)}$ . Indeed, recalling $x_n(a) = T_a^{n+1}(c)$ , just decompose

(3.3) $$ \begin{align} \xi_n(a)-\xi_n(a^{\prime})= \varphi_a(x_n(a))-\varphi_{a^{\prime}}(x_n(a)) +\varphi_{a^{\prime}}(T_a^{n+1}(c))- \varphi_{a^{\prime}}(T_{a^{\prime}}^{n+1}(c)). \end{align} $$

Fix $\alpha \in (0,1)$ such that (in view of the use of equation (2.30) in Corollary 3.4)

(3.4) $$ \begin{align} \frac{M_0}{1 - \alpha} \leq \frac{3}{\alpha}. \end{align} $$

Fix $q>1$ and $0<s<\operatorname \min \{\varpi , 1/q\}$ , and let

(3.5)

where $\unicode{x3bb} _\mathrm{CE}>1$ is given by equation (2.5), while $\rho =\operatorname \max \{\rho ^s_q, \rho _\varpi \}<1$ is given by Proposition 2.5, and $\theta \in (0,\operatorname \min \{1/2,\varpi \})$ is given by Lemma 2.8. Finally, recalling $\Lambda $ from equation (2.1), let $\eta \in (0,1/2)$ be so small that

(3.6)

Define the expectation $E(\psi )$ of $\psi \in L^\infty (\Omega _{*}^\varphi )$ by

(3.7) $$ \begin{align} E(\psi):= \frac{1}{m (\Omega_{*}^\varphi)} \int_{\Omega_{*}^\varphi} \psi \, dm. \end{align} $$

(We restrict to the Cantor set $\Omega _{*}^\varphi $ here and thus in equation (3.8). The bound in equation (2.9) is used in the proof of equation (3.8) (but not for equation (3.9), Lemma 3.3 or Corollary 3.4).) The following result is the key estimate on $\xi _j(a)=\varphi _a (T_a^{j+1}(c))$ .

Proposition 3.2. There exist $C_\varphi <\infty $ and $K<\infty $ such that

(3.8) $$ \begin{align} \bigg| E \bigg(\sum_{j=k}^{k+n-1} \xi_j \bigg)^2 -n \bigg| \leq C_\varphi \quad \text{for all } k \geq \operatorname\max \{2 K, [2/\eta]\}, \text{ for all } 1\leq n\leq \eta k/2, \end{align} $$

and, setting (the stretched exponent $1/4$ for $v(k)$ and the lower bound can be replaced by any number in $(0,1)$ , without changing the statements, up to adjusting intermediate constants) $v(k) = [k - k^{1/4}]$ for every non-trivial interval $\omega \subset \tilde \omega \in \mathcal {P}_{v(k)}$ with ${\omega \cap \Omega _{*}^\varphi \ne \emptyset} $ and $\unicode{x3bb} _0^{-k^{1/4}} \leq |x_{v(k)} (\omega )| \leq {v(k)}^{-3/\alpha }$ , we have

(3.9) $$ \begin{align} \bigg| \frac 1 {|\omega|} \int_\omega \bigg( \sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, dm - n \bigg| \leq C_\varphi \quad \text{for all } k \geq [2/\eta], \text{ for all } 1 \leq n \leq \eta k / 2 , \end{align} $$

and, for any sequence $\Psi _k$ with $C_\Psi :=\sup _k k^{-8/3}\sup |\Psi _k|<\infty $ and for any refinement $\mathcal {Q}_{v(k)}$ of $\mathcal {P}_{v(k)}$ such that $\unicode{x3bb} _0^{-k^{1/4}} \leq |x_v (\omega )| \leq v^{-3/\alpha }$ for all $\omega \in \mathcal {Q}_{v(k)}$ (we have $\unicode{x3bb} _0^{-k^{1/4}} \leq |x_v (\omega )|$ for all $\omega \in \mathcal {P}_{v(k)}$ by equation (3.21)), setting

(3.10) $$ \begin{align} \mathcal {Q}_{*,v(k)}:=\{\omega \in \mathcal {Q}_{v(k)} \mid \omega\cap \Omega_{*}^\varphi \ne \emptyset \} , \quad \Omega_{*,v(k)}^{\mathcal {Q}}=\bigcup_{\omega \in \mathcal {Q}_{*,v(k)}}\omega, \end{align} $$

we have

(3.11) $$ \begin{align} \bigg| E( \Psi_k) - \frac{1}{|\Omega_{*,v(k)}^{\mathcal {Q}}|} \int_{\Omega_{*,v(k)}^{\mathcal {Q}}} \Psi_k\, dm \bigg| \leq C_\Psi C_\varphi \quad \text{for all } k \geq [2/\eta]. \end{align} $$

Proposition 3.2 is proved in §3.1. Like for its analogue [Sch, Proposition 5.1], the first step will be to show the local estimate in equation (3.9) using Lemma 3.3 through its Corollary 3.4 (the analogues of [Sch, Lemma 5.3, Corollary 5.5]).

Lemma 3.3. (Switching locally from parameter to phase space)

Fix $\ell _0 \in \{1,2,3, 4\}$ . There exists $C<\infty $ such that we have, for any integers

$$ \begin{align*} n \leq n_i \leq n + \eta n, \quad 1 \leq i \leq \ell_0 , \end{align*} $$

for every $\tilde {\omega } \in \mathcal {P}_n$ and each non-trivial interval $\omega \subset \tilde {\omega }$ with $\omega \cap \Omega _{*}^\varphi \ne \emptyset $ ,

(3.12)

Corollary 3.4. There exists $C_3>1$ such that for $\ell _0$ , n, $ n_1, \ldots , n_{\ell _0}$ , and $\omega $ as in Lemma 3.3, if, in addition, $ |x_n (\omega )| \le n^{-3/\alpha }$ , then for any $a_0 \in \omega \cap \Omega _{*}^\varphi $ ,

Proof. Since equation (3.4) implies equation (2.30) for $\omega $ , the change of variables ${y = x_n (a)}$ on $\omega $ , combined with the distortion estimate in equation (2.31), gives

(3.13) $$ \begin{align} \nonumber &\bigg| \frac1{|\omega|} \int_{\omega } \prod_{\ell=1}^{\ell_0} \xi_{n_\ell}(a)\, da-\frac{1}{|x_n (\omega )|} \int_{x_n (\omega)} \prod_{\ell=1}^{\ell_0} \xi_{n_\ell}(x_n|_\omega^{-1} (y)) \, dy \bigg| \\ &\quad= \frac{1}{|x_n (\omega)|} \bigg| \int_{x_n (\omega)} \prod_{\ell=1}^{\ell_0} \xi_{n_\ell}(x_n|_\omega^{-1} (y)) \bigg( \frac 1{ |x_n^{\prime}(x_n|_\omega^{-1} y)|} \frac{|x_n (\omega )|}{|\omega |}-1\bigg ) dy \bigg| \\ \nonumber &\quad\leq C \frac{|x_n (\omega)|^\alpha}{|x_n (\omega )|} \int_{x_n (\omega)} \prod_{\ell=1}^{\ell_0} |\xi_{n_\ell}(x_n|_\omega^{-1}( y)) |\, dy. \end{align} $$

Since $\sup _k\|\xi _k\|_{L^\infty }<\infty $ , the claim then follows from Lemma 3.3.

Proof of Lemma 3.3

For $a_0\in \omega $ as in the statement, the functions

$$ \begin{align*} \tilde \varphi_\ell(y) =\tilde \varphi_{\ell,a_0}(y) = \varphi_{a_0} (T_{a_0}^{n_\ell - n} (y)) , \quad \tilde \xi_\ell (y)=\tilde \xi_{\ell,\omega} (y)= \xi_{n_\ell} (x_n|_\omega^{-1} (y)) \end{align*} $$

with

$$ \begin{align*} \xi_{n_\ell} (x_n|_\omega^{-1} (y)) =\varphi_{x_n|_\omega^{-1} (y)} (x_{n_\ell} (x_n|_\omega^{-1} (y))= \varphi_{x_n|_\omega^{-1} (y)} (T^{n_\ell+1}_{x_n|_\omega^{-1} (y)}(c)) \end{align*} $$

are bounded on $x_n(\omega \cap \Omega ^\varphi _{*})$ . Decomposing

(3.14) $$ \begin{align} \nonumber |\tilde \xi_1 \tilde \xi_2 \tilde \xi_3 \tilde \xi_4 - \tilde \varphi_1 \tilde \varphi_2 \tilde \varphi_3 \tilde \varphi_4| &\leq |(\tilde \xi_1 -\tilde \varphi_1) \tilde \xi_2 \tilde \xi_3 \tilde \xi_4| + |\tilde \varphi_1 (\tilde \xi_2 - \tilde \varphi_2) \tilde \xi_3 \tilde \xi_4| \\ &\quad+ |\tilde \varphi_1 \tilde \varphi_2 (\tilde \xi_3 - \tilde \varphi_3) \tilde \xi_4| + |\tilde \varphi_1 \tilde \varphi_2 \tilde \varphi_3 (\tilde \xi_4 - \tilde \varphi_4) | , \end{align} $$

it is enough to find a uniform constant $\bar C>1$ such that

We will do so by showing the pointwise estimate

For $a= x_n|_\omega ^{-1} (y)$ , we decompose

(3.15) $$ \begin{align} \nonumber &\tilde \xi_{\ell,\omega} (y) - \tilde \varphi_{\ell,a_0} (y) = \xi_{n_\ell} (a) - \varphi_{a_0} (T_{a_0}^{n_\ell - n} (x_n (a))) \\ \nonumber &\quad= \varphi_a(x_{n_\ell} (a)) - \varphi_{a_0} (T_{a_0}^{n_\ell - n} (x_n (a)))\\ &\quad= \varphi_a(x_{n_\ell} (a)) - \varphi_{a_0} (x_{n_\ell} (a)) + \varphi_{a_0} (x_{n_\ell} (a)) - \varphi_{a_0} (T_{a_0}^{n_\ell - n} (x_n (a))). \end{align} $$

Using Remark 3.1, there exists C, independent of $n_\ell $ , such that

(3.16) $$ \begin{align} |\varphi_{a} (x_{n_\ell} (a))- \varphi_{a_0} (x_{n_\ell} (a)) | \leq C |\omega|^\theta \quad \text{for all } \{a, a_0\}\subset \omega. \end{align} $$

Hence, using our choice in equation (3.5) of $\unicode{x3bb} _0$ , and since $|\omega | \leq C \unicode{x3bb} _\mathrm{CE}^{-n}$ by equation (2.7), we get

(3.17)

For the last two terms in the right-hand side of equation (3.15), note that since $a=x_n|_\omega ^{-1} (y)$ implies $x_{n_\ell }(a) = T^{n_\ell + 1}_a(c) = T^{n_\ell -n}_a (T^{n+1}_a (c)) = T^{n_\ell -n}_a(y)$ , we have, using equation (2.2),

(3.18) $$ \begin{align} &| x_{n_\ell} (x_n|_\omega^{-1} (y)) - T_{a_0}^{n_\ell - n}(y) | =|T_a^{n_\ell - n} (y) - T_{a_0}^{n_\ell - n} (y)| \nonumber\\ &\quad\leq C \Lambda^{n_\ell - n} |a - a_0| \leq C \Lambda^{n_\ell - n}|\omega| \quad \text{for all } y \in x_n(\omega). \end{align} $$

Then, since $n_\ell -n\le \eta n$ , our choice of $\unicode{x3bb} _0$ , $\eta $ , with equation (3.2) at $a=a_0$ give (we do not need the analogue of [Sch, Sublemma 5.4] here)

(3.19)

using again in the last inequality that $|\omega | \leq C \unicode{x3bb} _\mathrm{CE}^{-n}$ from equation (2.7). We conclude by combining equations (3.17) and (3.19) into equation (3.15).

3.1 Proof of Proposition 3.2

We first show equation (3.9). Let $\omega \subset \tilde \omega \in \mathcal {P}_{[k-k^{1/4}]}$ , with $k\ge 2n/\eta $ , be as in the assertion. Writing

$$ \begin{align*} \int_{\omega} \bigg( \sum_{j=k}^{k+n-1}\xi_j \bigg)^2 \, dm = \sum_{j=k}^{k+n-1} \bigg( \int_{\omega}\xi_j^2 \, dm + 2 \sum_{\ell=j+1}^{k+n-1} \int_{\omega }\xi_j \xi_\ell \, dm \bigg), \end{align*} $$

it is sufficient to show that

(3.20) $$ \begin{align} \sum_{j=k}^{k+n-1} \bigg| 1 - \frac1{|\omega|} \int_{\omega} \bigg( \xi_j^2 + 2 \sum_{\ell=j+1}^{k+n-1} \xi_j \xi_\ell \bigg) \, dm \bigg| = O(1). \end{align} $$

Fix $a_0 \in \omega \cap \Omega _{*}^\varphi $ . By Corollary 3.4 for $\ell _0=2$ , we have, for $k\le j \le k+n-1$ ,

(recall $v=[k-k^{1/4}]$ ). Since $0<s<1/q<1$ , we have that ${1}_{x_v(\omega )} \in H^s_q$ , uniformly in v and $\omega $ (see [St]), so the first claim of Proposition 2.5 gives

$$ \begin{align*} &\int_{x_v(\omega)} (\varphi_{a_0} \circ T_{a_0}^{j-v} ) (\varphi_{a_0} \circ T_{a_0}^{\ell-v} )\, dm \\ &\quad = |x_v(\omega)| \int \varphi_{a_0} \cdot (\varphi_{a_0} \circ T_{a_0}^{\ell-j}) \, d\mu_{a_0} + O(\rho^{j-v}) \quad \text{for all } \ell \geq j. \end{align*} $$

Hence,

By equations (1.3) and (1.7), we have

$$ \begin{align*} 1 = \int \varphi_{a_0}^2 \, d \mu_{a_0} + 2 \sum_{i=1}^\infty \int \varphi_{a_0} \cdot \varphi_{a_0} \circ T_{a_0}^i \, d \mu_{a_0}. \end{align*} $$

Therefore, the second claim of Proposition 2.5 gives

$$ \begin{align*} \int \bigg( \varphi_{a_0}^2 + 2 \sum_{\ell=j+1}^{k+n-1} \varphi_{a_0} \cdot (\varphi_{a_0} \circ T_{a_0}^{\ell-j}) \bigg) \, d\mu_{a_0} = 1 + O (\rho^{k+n-j}).\end{align*} $$

Hence, we find, for $k \leq j \leq k + n - 1$ and $v = [k - k^{1/4}]$ ,

To proceed, we shall use several times that

$$ \begin{align*} \sup_{n}\sup_k \sum_{j=k}^{k+n-1} \frac{1}{(k+n-j)^2} \le \sup_n \sum_{\ell=1}^n \frac 1 {\ell^2} <\infty. \end{align*} $$

Clearly, $\rho ^{k+n-j} \leq {C}/{(k+n-j)^2}$ . For the term $(k+n-j) \rho ^{j-v} |x_v (\omega )|^{-1}$ , we use $|x_v (\omega )| \geq \unicode{x3bb} _0^{-k^{1/4}}$ and the definition in equation (3.5) of $\unicode{x3bb} _0$ to get, since $k\ge 2n/\eta $ ,

The term $(k+n-j) \unicode{x3bb} _0^{-(k-k^{1/4})}$ is similar. Finally, $|x_v (\omega )| \leq v^{-3/\alpha }$ gives

$$ \begin{align*} \sum_{j=k}^{k+n-1} (k + n - j) |x_v (\omega)|^\alpha \leq \sum_{j=k}^{k+n-1} \frac{k+n-j}{n^3}\leq n \frac{k+n-k}{n^3} = \frac{1}{n}. \end{align*} $$

This proves equation (3.20), and hence equation (3.9).

We will next deduce equations (3.8) and (3.11) from equation (3.9). Fix $\kappa _1>\kappa _0$ , let $N_1(\kappa _1)\ge N_0$ be given by Lemma 2.3 and let $K\ge N_1$ be such that $ k^{\kappa _1}\le \unicode{x3bb} _0^{k^{1/4}}$ for all $k \ge K$ . Then, if $v = v(k) \ge K$ (so that $k\ge K$ ), we have

(3.21)

Refining $\mathcal {P}_v$ to a partition $\mathcal {Q}_v$ such that

we set $\Omega _{*,v}^{\mathcal {Q}}$ as in equation (3.10) and we decompose

$$ \begin{align*} |\Omega^\varphi_{*}| \cdot E \bigg(\sum_{j=k}^{k+n-1} \xi_j \bigg)^2 &= \int_{\Omega^{\mathcal {Q}}_{*,v}} \bigg( \sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m - \int_{\Omega^{\mathcal {Q}}_{*,v}\setminus \Omega^\varphi_{*}} \bigg( \sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m \\ &= \int_{\Omega_v} \bigg( \sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m - \int_{\Omega_v\setminus \Omega^\varphi_{*}} \bigg( \sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m. \end{align*} $$

Then, using equation (2.9), $\sup _k \sup |\xi _k|<\infty $ , $\kappa _0 \geq 3 / d_1$ and $v(k)\ge k/2\ge n/\eta $ ,

(3.22) $$ \begin{align} \nonumber 0 \leq \int_{\Omega^{\mathcal {Q}}_{*,v}\setminus \Omega^\varphi_{*}} \bigg (\sum_{j=k}^{k+n-1} \xi_j \bigg)^2\, d m & \leq \int_{\Omega_v\setminus \Omega^\varphi_{*}} \bigg (\sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m \\ & \leq C n^2 e_v \leq C n^2 n^{1 - d_1 \kappa_0} \leq C , \end{align} $$

which shows that

$$ \begin{align*} \int_{\Omega_{*}^\varphi} \bigg (\sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m & = \int_{\Omega^{\mathcal {Q}}_{*,v}} \bigg( \sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m + O(1) \\ &= \int_{\Omega_v} \bigg( \sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m + O(1). \end{align*} $$

By equation (3.9),

(3.23) $$ \begin{align} \int_{\omega} \bigg (\sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m = |\omega|(n + O(1)) \quad \text{for all } \omega \in \mathcal {Q}_v^*. \end{align} $$

Summing equation (3.23) over $\omega \in \mathcal {Q}^*_v$ , we get that

$$ \begin{align*} \int_{\Omega^{\mathcal {Q}}_{*,v}} \bigg (\sum_{j=k}^{k+n-1} \xi_j \bigg)^2 \, d m = n + O(1). \end{align*} $$

Finally, using again equation (2.9) to see

$$ \begin{align*} 0 \leq \frac{|\Omega^{\mathcal {Q}}_{*,v}|}{|\Omega^\varphi_{*}|} - 1 \leq \frac{|\Omega_v|}{|\Omega^\varphi_{*}|}-1 = O(e_v) = O(e_n), \end{align*} $$

we have established equation (3.8), and also equation (3.11) in the case ${\Psi _k\kern1.5pt{=}\kern1.5pt(\sum _{j=k}^{k+n-1} \xi _j )^2}$ (note that $|\Psi _k|\le C n^2 \le C k^2$ ). For more general $\Psi _k$ , the same argument, using ${d_1 \kappa _0 \ge 11/3}$ in equation (3.22), gives equation (3.11). This ends the proof of Proposition 3.2.

4 Proof of Theorem 1.1 via Skorokhod’s representation theorem

We will rearrange the Birkhoff sum as a sum of blocks of polynomial size, approximate the blocks by a martingale and finally apply Skorokhod’s representation theorem to this martingale. The size for the jth block $\mathbb {I}_j$ is $j^{2/3}$ , which will give the error exponent $\gamma>2/5$ in our ASIP. (A block size $\#\mathbb {I}_j=j^b$ replaces $3/5$ in equation (4.4) by $1/(1+b)$ , so that the first constraint becomes $N^\gamma> N^{b/(1+b)}$ , see equation (4.5). Our bounds in equations (4.25) and (4.26) (with Gál–Koksma’s strong law of large numbers, Theorem 4.3, and $M(N)\sim N^{1/(1+b)}$ ) give $N^{\gamma }> N^{(b+2)/(4(b+1))}$ . Hence, $b=2/3$ is the optimum. In the independent and identically distributed case, a block size $j^{1/2}$ gives $\gamma>1/3$ [PS, p. 25], see also the beginning of [Sch, §6].)

4.1 Blocks $\mathbb {I}_M$ . Approximations $\chi _i$ and $y_j$

Fix $a_{*}$ , $\varpi \in (0,1)$ , q, $s\in (0, \operatorname \min \{\varpi , 1/q\})$ , $\rho $ , $\theta $ , $\unicode{x3bb} _0$ , $\eta $ , $\alpha $ , $\varphi \in C^\varpi $ , $\Omega _{*}^\varphi =\Omega _{*}\cap [a_{*}-\epsilon _\varphi , a_{*}+\epsilon _\varphi ]$ as in the beginning of §3. Set

$$ \begin{align*} \mathcal{P}_{*,k}:=\{\omega \in \mathcal{P}_{k} \mid |\omega\cap \Omega_{*}^\varphi|>0\} , \quad \Omega_{*, k}:= \bigcup_{\omega \in \mathcal{P}_{*, k}} \omega, \quad k \ge 1. \end{align*} $$

Fix $\gamma \in (2/5, 1/2)$ and $\delta \in (0,\operatorname \min \{1/5, 2(\gamma -2/5)\})$ . (See Lemma 4.4 for the condition $\delta <2(\gamma -2/5)$ .) For $i\ge 1$ , we shall approach $\xi _i \colon [a_{*}-\epsilon _\varphi ,a_{*}+\epsilon _\varphi ] \to \mathbb {C}$ (see Remark 3.1) by the stepfunction

$$ \begin{align*} \chi_i\colon \Omega_{*, r_i} \to \mathbb{C} , \quad \chi_i = E(\xi_i | \mathcal{F}_{r_i}) \quad \mbox{where } r_i=i+[i^{\delta}] , \end{align*} $$

with $\mathcal {F}_k$ the $\sigma $ -algebra generated by the intervals in $\mathcal {P}_{*,k}$ . Conditional expectations are only defined almost everywhere, but we may set (see equation (3.7))

(4.1) $$ \begin{align} \chi_i|_{\omega} \equiv \frac{ \int_{\omega \cap \Omega_{*}^\varphi} \xi_i \, dm}{|\omega \cap \Omega_{*}^\varphi|} \quad \text{for all } \omega \in \mathcal{P}_{*,r_i} , \text{ for all } i \ge 1. \end{align} $$

Thus, $\chi _i$ is defined everywhere on $\Omega _{*,r_i}$ , allowing pointwise claims about it.

Recalling $e_\ell $ from equation (2.9) and our assumption $\unicode{x3bb} _\mathrm{CE}>e^{ 14 \alpha _\mathrm{BC}}$ in the proof of Proposition 2.2, we have the following basic lemma.

Lemma 4.1. For any $\tilde{\unicode{x3bb}} _\mathrm{CE}\in (e^{\alpha _\mathrm{BC}}, \sqrt {\unicode{x3bb} _\mathrm{CE} } \cdot e^{-\alpha _\mathrm{BC}})$ , there exists C such that

(4.2)

and for all $i\ge 1$ , $j\ge 0$ and all $a\in \Omega _{*,r_i}$ ,

(4.3) $$ \begin{align} |E(\xi_{i+j} | \mathcal{F}_{r_i} )(a)| = |E(\chi_{i+j} | \mathcal{F}_{r_i})(a)| \leq C \operatorname\min (1, e_{ [\eta (j - 2 i^\delta)]}). \end{align} $$

(The constant C in equation (4.3) goes to infinity as $\delta \to 0$ , that is, if $\gamma \to 2/5$ .)

Following [PS, §3.3], [Sch, §6.1], we define inductively consecutive blocks $\mathbb {I}_j$ of integers and associated functions $y_j$ as follows. Let $\mathbb {I}_1 = \{1\}$ and let $\mathbb {I}_j$ for $j\ge 2$ contain $[j^{2/3}]$ consecutive integers. The first blocks are

$$ \begin{align*} \underbrace{1,}_{\mathbb{I}_1} \underbrace{2,}_{\mathbb{I}_2} \underbrace{3, 4,}_{\mathbb{I}_3} \underbrace{5, 6,}_{\mathbb{I}_4} \underbrace{7, 8,}_{\mathbb{I}_5} \underbrace{9, 10, 11,}_{\mathbb{I}_6} \underbrace{12, 13, 14,}_{\mathbb{I}_7} \underbrace{15, 16, 17, 18,}_{\mathbb{I}_8} \underbrace{19, 20, 21, 22,}_{\mathbb{I}_9} \ldots \end{align*} $$

Let $M = M(N)$ be uniquely defined by $N \in \mathbb {I}_M$ . There exists C such that

(4.4) $$ \begin{align} C^{-1} N^{3/5} \leq M(N) \leq C N^{3/5} \quad \text{for all } N \geq 1. \end{align} $$

By equation (4.2) in Lemma 4.1, there is C such that, for all $i\ge 1$ and all $a \in \Omega _{*, r_i}$ ,

(4.5) $$ \begin{align} \bigg| \sum_{i=1}^N\xi_i(a) - \sum_{j=1}^{M(N)} \sum_{i \in \mathbb{I}_j} \chi_i (a) \bigg| & \le\sum_{i=1}^N|\xi_i(a)-\chi_i(a)|+C\# \mathbb{I}_M \le C N^{2/5} \end{align} $$

for all $N\ge 1$ . Hence, to prove Theorem 1.1, it is sufficient to consider

$$ \begin{align*} y_j \colon \Omega_{*, [Cr_j^{5/3}]} \to \mathbb{C}, \quad y_j := \sum_{i \in \mathbb{I}_j} \chi_i , \quad j \ge 1. \end{align*} $$

Proof of Lemma 4.1

By equation (4.1), since $\xi _i$ is continuous (see Remark 3.1), for any $\omega \in \mathcal {P}_{*,r_i}$ , there exists $a^{\prime }\in \omega $ such that $\chi _i|_{\omega }=\xi _i(a^{\prime })$ . Revisiting the decomposition in equation (3.3), and using equation (3.2) and the $\theta $ -Hölder continuity of $a\mapsto \varphi _a(u)$ (as for equation (3.16)), we find C such that for all $i\ge 1$ and $\omega \in \mathcal {P}_{*,r_i}$ ,

$$ \begin{align*} |\xi_i(a)-\chi_i(a)|= |\xi_i (a) - \xi_i (a^{\prime})| & \leq C ( |\omega|^\theta+ |x_i (\omega)|^\varpi ) \leq C |x_i (\omega)|^{\theta } \quad \text{ for all } a \in \omega, \end{align*} $$

where we used $\theta \le \varpi $ and equation (2.7) in the second inequality. This establishes equation (4.2), since for any $\bar{\unicode{x3bb}} _\mathrm{CE}\in (e^{\alpha _\mathrm{BC}}, \sqrt {\unicode{x3bb} _\mathrm{CE} } \cdot e^{-\alpha _\mathrm{BC}})$ , there exists $\bar C$ such that

(4.6)

To show equation (4.6), first note, using equation (2.29), that there exists $a\in \omega $ such that

$$ \begin{align*} |x_i(\omega)| \leq C \frac{|x_{r_i}(\omega)|}{|(T_a^{i^\delta})^{\prime}(x_i(a))|}. \end{align*} $$

Then, if $a\in \Omega _{*}$ , the polynomial recurrence in equation (2.8) and standard arguments give

(4.7)

(see e.g. [BS1, Proposition 3.7] in the exponentially recurrent case). If $a\notin \Omega _{*}$ , we may use bounded distortion in equation (2.31) ( $\alpha =0$ suffices here) since $|\omega \cap \Omega _{*}|>0$ .

The equality in equation (4.3) follows from the definition since $\mathcal {F}_{r_i} \subset \mathcal {F}_{r_{i+j}}$ . Indeed, for $a \in \omega \in \mathcal {P}_{*,r_i}$ ,

(4.8) $$ \begin{align} |\omega \cap \Omega_{*}^\varphi|\cdot |E(\xi_{i+j} | \mathcal{F}_{r_i} )(a)|&= \int_{\omega \cap \Omega_{*}^\varphi} \xi_{i+j} \, dm\nonumber\\ & =\sum_{\substack{\omega^{\prime} \in \mathcal{P}_{*,r_{i+j}}\\ \omega^{\prime}\subset \omega}} |\omega^{\prime} \cap \Omega_{*}^{\varphi}| \cdot \frac{ \int_{\omega^{\prime} \cap \Omega_{*}^{\varphi}} \xi_{i+j} \, dm}{|\omega^{\prime} \cap \Omega_{*}^{\varphi}|}\nonumber\\ &=\sum_{\substack{\omega^{\prime} \in \mathcal{P}_{*,r_{i+j}}\\ \omega^{\prime}\subset \omega}}|\omega^{\prime} \cap \Omega_{*}^{\varphi}| \cdot \chi_{i+j}|_{\omega^{\prime}} =\sum_{\substack{\omega^{\prime} \in \mathcal{P}_{*,r_{i+j}}\\ \omega^{\prime}\subset \omega}} \int_{\omega^{\prime}\cap \Omega_{*}^{\varphi}} \chi_{i+j}\, dm. \end{align} $$

Since $\sup _k \|\xi _k\|_{L^\infty }<\infty $ , we may and shall assume that $j \geq 2 i^{\delta }$ to prove the upper bound in equation (4.3). For such j, recalling $\eta \in (0,1/2)$ from equation (3.6), define

(4.9) $$ \begin{align} k=k(i,j)= \operatorname\max \bigg\{ i + [i^\delta] + \eta (j - i^\delta), \bigg \lceil \frac{i+j}{1 + \eta}\bigg \rceil \bigg\} \end{align} $$

so that $k\le i+j- \eta /({1+\eta })(j-i^\delta ) \le i+j$ and $i+j \le k(1+\eta )$ .

Since $\delta $ is fixed, we may and shall assume that i is large enough such that $k(i,j)\ge N_1$ (with $N_1$ from Lemma 2.3) and

(4.10)

Since $k(i,j) \geq r_i$ , we have, similarly as for equation (4.8),

$$ \begin{align*} |E(\xi_{i+j} | \mathcal{F}_{r_i}) (a)| = |E(E(\xi_{i+j} | \mathcal{F}_{k(i,j)}) | \mathcal{F}_{r_i})(a)| \quad \text{for all } a \in \tilde \omega\in \mathcal{P}_{*, r_i}. \end{align*} $$

We must analyse the above decomposition more closely than in the proof of [Sch, Lemma 6.1] as follows. Let $a \in \tilde \omega \in \mathcal {P}_{*, r_i}$ , then,

(4.11) $$ \begin{align} &|\tilde \omega\cap \Omega_{*}^\varphi| \cdot |E(E(\xi_{i+j} | \mathcal{F}_{k(i,j)}) | \mathcal{F}_{r_i})(a)| = \bigg | \sum_{\substack{\omega \in \mathcal{P}_{*,k(i,j)}\nonumber\\ \omega \subset \tilde \omega}} \frac {|\omega\cap \Omega^\varphi_{*}|} {|\omega\cap \Omega^\varphi_{*}|} \int_{\omega\cap \Omega_{*}^\varphi} \xi_{i+j} \, d m\bigg | \\ &\quad\le \bigg |\sum_{\substack{\omega \in \mathcal{P}_{*,k(i,j)}\\\omega \subset \tilde \omega}} \frac {|\omega|} {|\omega|} \int_{\omega} \xi_{i+j} \, d m\bigg | + \sup_{\tilde a} \|\varphi_{\tilde a}\|_{L^\infty}\cdot \sum_{\substack{\omega\in \mathcal{P}_{*,k(i,j)}\\ \omega \subset \tilde \omega}} |\omega \setminus (\omega\cap \Omega^\varphi_{*})|. \end{align} $$

Since $\tilde \omega \in \mathcal {P}_{*, r_i}$ , the bound in equation (2.10) implies

(4.12) $$ \begin{align} \begin{cases} \dfrac{\sum_{\substack{\omega\in \mathcal{P}_{*,k(i,j)}\\ \omega\subset \tilde \omega}} |\omega \setminus (\omega\cap \Omega_{*})| }{|\tilde \omega\cap \Omega_{*}^\varphi|} \le \dfrac{d_0 e_{k(i,j)-r_i} | \tilde \omega|} {(1-d_0 e_{r_i}) | \tilde\omega|} \le C d_0 e_{[\eta(j-i^\delta)]}, \\[6pt] |\tilde \omega | /{|\tilde \omega\cap \Omega_{*}^\varphi|} \le \dfrac{ | \tilde \omega|} {(1-d_0 e_{r_i})| \tilde \omega|}\le C. \end{cases} \end{align} $$

In view of equations (2.10), (4.12) and (4.11), it suffices to show

$$ \begin{align*} \frac1{|\omega|} \bigg| \int_{\omega} \xi_{i+j} \, dm \bigg| \le C \operatorname\min (1, e_{[\eta (j- 2 i^\delta)]}) \quad \text{for all } \omega \in \mathcal{P}_{*,k(i,j)}. \end{align*} $$

Fix $\omega \in \mathcal {P}_{*,k(i,j)}$ . First note that, by equation (2.31) for $\alpha =0$ ,

(4.13) $$ \begin{align} \frac1{|\omega|} \bigg| \int_\omega \xi_{i+j}(a) \, da \bigg| \leq \frac{C}{|x_k(\omega)|} \bigg| \int_{x_k(\omega)} \xi_{i+j} (x_k|_\omega^{-1}(y)) \, dy \bigg|. \end{align} $$

Then, on the one hand, Lemma 3.3 for $\ell _0=1$ gives $a_0\in \omega \cap \Omega ^\varphi _{*}$ such that

(4.14)

On the other hand, recalling $0<s<1/q$ , since $1_{x_k (\omega )}\in H^s_q$ (uniformly in k and $\omega $ ), the first claim of Proposition 2.5, with $\int \varphi _{a_0} d\mu _{a_0}=0$ , gives (a factor $|x_k (\omega )|^{-1}$ was omitted when applying [Sch], Proposition 4.3] on [Sch, p. 400]. We fix this by using our polynomial lower bound on $|x_k(\omega )|$ (considering two different values of $\delta $ should work for [Sch]))

(4.15) $$ \begin{align} &\frac1{|x_{k}(\omega)|} \bigg| \int_{x_{k}(\omega)} \varphi_{a_0}(T_{a_0}^{i+j-k}(y)) \,dy \bigg| \leq C \cdot {k(i,j)^{(\kappa_0+1)}} \rho^{i+j-k(i,j)} \nonumber \\ &\quad \leq C \cdot (i+j)^{\kappa_0+1} \rho^{\eta(j-i^\delta)/(1+\eta)} \leq C \cdot (2j)^{(\kappa_0+1)/\delta} \rho^{\eta j/(2+2\eta)}. \end{align} $$

(We used $|x_{k}(\omega )|> Ck^{-\kappa _0+1}$ from Lemma 2.3.) Putting together equations (4.13), (4.14), (4.15) and (4.10), we conclude the proof of equation (4.3).

4.2 Law of large numbers for $y_j^2$

Recall that $\gamma \in (2/5,1/2)$ is fixed. The main ingredient in the proof of Theorem 1.1 is the following analogue of [Sch, Lemma 6.2], itself inspired by [PS, Lemma 3.3.1].

Lemma 4.2. For $m_{*}$ -a.e. $a \in \Omega _{*}^\varphi $ , there exists $C(a)$ such that

(4.16) $$ \begin{align} \bigg| N - \sum_{j=1}^{M(N)}y_j^2(a) \bigg| \leq C(a) N^{2\gamma} \quad \text{for all } N \geq 1. \end{align} $$

The proof of Lemma 4.2 (which uses Proposition 3.2 and equation (4.2), but not equation (4.3)) is based on the following theorem ([GK], see also [PS, Theorem A.1]).

Theorem 4.3. (Gál–Koksma’s strong law of large numbers)

Let $z_j$ , $j\ge 1$ , be zero-mean random variables. Assume there exist $p\ge 1$ and $C<\infty $ with

$$ \begin{align*} E \bigg( \sum_{j = m + 1}^{m + n} z_j \bigg)^2 \leq C((m + n)^p - m^p) \quad \text{for all }\ m \geq 0\ \text{and}\ n \geq 1. \end{align*} $$

Then, for all $\iota>0$ , we have $(1/{n^{p/2 + \iota }}) \sum _{j = 1}^n z_j \to 0$ almost surely.

Proof of Lemma 4.2

Set $w_j = \sum _{i \in \mathbb {I}_j} \xi _i$ . Since $y_j^2 - w_j^2 = (y_j + w_j) (y_j - w_j)$ and $|y_j + w_j|\leq C j^{2/3}$ , the bound in equation (4.2) gives C such that $|y_j^2 - w_j^2| \leq C j^{2/3} \tilde{\unicode{x3bb}} _\mathrm{CE}^{-\theta j^\delta }$ for all $j\ge 1$ and $a \in \Omega _{*, r_{Cj^{5/3}}}$ . Hence, $\sup _{a\in \Omega _{*}^\varphi }\sum _{j \geq 1} |y_j^2 - w_j^2|$ is finite, and it suffices to show equation (4.16) with $y_j$ replaced by $w_j$ .

By equation (3.8), we have $| E (w_j^2) - \# \mathbb {I}_j | \leq C$ , and, since $\sum _{j=1}^{M(N)} \# \mathbb {I}_j = N$ , we get $ | \sum _{j=1}^{M(N)} {E( w_j^2 )- N | \leq C M(N)}$ . Therefore,

(4.17) $$ \begin{align} \bigg| N - \sum_{j=1}^{M(N)} w_j^2 \bigg| \leq CM(N) + \bigg| \sum_{j=1}^{M(N)} w_j^2 - E( w_j^2 )\bigg|. \end{align} $$

Assume there exists C such that

(4.18) $$ \begin{align} E \bigg( \sum_{j = m + 1}^{m + n} w_j^2 - E (w_j^2) \bigg)^2 \leq C( (m + n)^{8/3 } -m^{8/3}) \quad \text{for all } m \geq 0,\ n \geq 1. \end{align} $$

Then, Theorem 4.3 (Gál–Koksma) applied to $\iota \in (0, 10(\gamma -2/5)/3]$ , $p=8/3$ and the zero-mean random variables $z_j=w_j^2-E( w_j^2)$ , implies that

$$ \begin{align*} \sum_{j = 1}^{M(N)} w_j^2 - E (w_j^2) = o (M^{{4}/{3} + \iota}) \quad \mbox{almost surely}. \end{align*} $$

Hence, equation (4.17) gives $ | N - \sum _{j=1}^{M(N)} w_j^2(a) | \leq C(a) N^{4/5 + 3\iota /5}\le C(a)N^{2\gamma } $ , almost surely (recall $M(N) \sim N^{3/5}$ by equation (4.4)). It remains to prove equation (4.18).

By Jensen’s inequality, we have $(E(w_j^2))^2 \leq E(w_j^4)$ and, therefore,

(4.19) $$ \begin{align} E \bigg(\sum_{j = m + 1}^{m + n} w_j^2 - E (w_j^2) \bigg)^2 \leq 2 \sum_{j = m + 1}^{m + n} \bigg( E (w_j^4) + \sum_{k = j + 1}^{m + n} |E (w_j^2 w_k^2) - E (w_j^2) E (w_k^2)| \bigg). \end{align} $$

We consider first $E(w_j^4)$ . Fix $\upsilon \in (0,1/6)$ and, for $j\ge 1$ , let

$$ \begin{align*} S_j = \{\, \vec v \in \mathbb{I}_j^4 \mid v_1 \leq v_2 \leq v_3 \leq v_4 \mbox{ and } \operatorname{\max}\{v_2 - v_1 , v_4-v_3\}\ge j^\upsilon \,\}. \end{align*} $$

Then, since $\#( \{\, \vec v \in \mathbb {I}_j^4 \mid v_1 \leq v_2 \leq v_3 \leq v_4\} \setminus S_j)\le (j^{2/3 + \upsilon })^2 = j^{4/3+2\upsilon }$ , we find

(4.20) $$ \begin{align} \int\limits_{\Omega_{*}^\varphi} w_j(a)^4 \, da &= \sum_{\vec v \in \mathbb{I}_j} \bigg| \int_{\Omega_{*}^\varphi} \prod_{\ell = 1}^4 \xi_{v_\ell} (a) \, da \bigg| \nonumber \leq C \sum_{\substack{\vec v \in \mathbb{I}_j^4 \\ v_1 \leq \cdots \leq v_4}} \bigg| \int_{\Omega_{*}^\varphi } \prod_{\ell=1}^4 \xi_{v_\ell} (a) \, da \bigg| \nonumber \\ & \leq C \sum_{\vec v\in S_j} \bigg| \int_{\Omega_{*}^\varphi} \prod_{\ell=1}^4\xi_{v_\ell}(a) \, da \bigg| + C j^{4/3+2\upsilon}. \end{align} $$

Let $\vec v\in S_j$ be such that $v_4-v_3 \geq j^\upsilon $ . For $\omega \in \mathcal {P}_{v_3}$ such that $\omega \cap \Omega _{*}^\varphi \ne \emptyset $ , the change of variable in equation (3.13), together with an easy variant of Lemma 3.3 deduced from equation (3.14), give $a_0\in \omega \cap \Omega _{*}^\varphi $ such that

For $y\in x_{v_3}(\omega )$ , setting $a = x_{v_3}|_\omega ^{-1} (y)$ , and recalling Remark 3.1, we find

$$ \begin{align*} &|\xi_{v_\ell} (x_{v_3}|_\omega^{-1} (y)) - \varphi_{a_0} (x_{v_\ell} \circ x_{v_3}|_\omega^{-1}(y))| \\ &\quad = |\varphi_a (x_{v_\ell} \circ x_{v_3}|_\omega^{-1}(y)) - \varphi_{a_0} (x_{v_\ell} \circ x_{v_3}|_\omega^{-1}(y))| \leq C |\omega|^\theta \end{align*} $$

for $\ell =1, 2, 3$ . Thus, equations (2.9) and (2.7) imply (using $\sup _k\|\xi _k\|_{L^\infty }<\infty $ )

(4.21)

We claim that for $\ell =1,2,3$ , and for each $\omega \in \mathcal {P}_{*, v_3}$ ,

(4.22) $$ \begin{align} |\partial_y(x_{v_\ell} \circ( x_{v_3}|_\omega^{-1}))(y)| \leq C v_{\ell}^{\kappa_0} \quad \text{for all } y \in x_{v_3}(\omega). \end{align} $$

Indeed, by equation (2.29), there exists $a \in \omega $ such that

$$ \begin{align*} |\partial_y (x_{v_\ell} \circ( x_{v_3}|_\omega^{-1}))(y)|\le C |(T_a^{v_3-v_\ell})^{\prime}(T_a^{v_\ell+1}(c))|^{-1}. \end{align*} $$

Thus, if $a\in \Omega _{*}$ , standard arguments (see e.g. [BS1, Proposition 3.7], using our polynomial recurrence in equation (2.8)) give the claim. Otherwise, since $|\omega \cap \Omega _{*}|>0$ , we may use equation (2.31) as for equation (4.6).

Therefore, we find C such that for each $v_3$ and $\omega \in \mathcal {P}_{*, v_3}$ ,

$$ \begin{align*} \| 1_{x_{v_3} (\omega)} \cdot \prod_{\ell=1}^3(\varphi_{a_0} \circ x_{v_\ell} \circ x_{v_3}|_\omega^{-1}) \|_{H^s_q} \leq C (v_1 v_2)^{\varpi \kappa_0}\| \varphi_{a_0} \|^2_{C^\varpi} \| \varphi_{a_0} \|_{H^s_q}. \end{align*} $$

Indeed, on the one hand, there exists C such that, for any $C^2$ map $\mathcal {T}$ , we have

$$ \begin{align*} \|\varphi_{a_0}\circ \mathcal{T} \|_{C^\varpi}\le C \sup|\mathcal{T}^{\prime}|^\varpi \|\varphi_{a_0}\|_{C^\varpi}. \end{align*} $$

On the other hand, since $0<s<1/q<1$ , the characteristic function of an interval is a bounded multiplier on $H^s_q(I)$ (uniformly in the size of the interval), and since $s<\varpi $ , a function in $C^\varpi $ is a bounded multiplier on $H^s_q(I)$ [St, Th].

Hence, by the first claim of Proposition 2.5 (with equation (3.2) and $\int \varphi _{a_0} d\mu _{a_0}=0$ ), we have

$$ \begin{align*} \frac{| \int_{x_{v_3}(\omega)} ( \prod_{\ell=1}^3 \varphi_{a_0}(x_{v_\ell} \circ x_{v_3}|_\omega^{-1}) ) \varphi_{a_0}(T_{a_0}^{v_4-v_3}) \, dy |}{|x_{v_3}(\omega)|} &\le C (v_1 v_2)^{\varpi \kappa_0} \frac{\rho^{v_4-v_3}}{|x_{v_3}(\omega)|}\\ &\le C j^{10\varpi \kappa_0/3} v_3^{\kappa_1}\rho^{j^\upsilon}. \end{align*} $$

(We used Lemma 2.3 and that $v_\ell \in \mathbb {I}_j$ implies $v_\ell \leq C j^{5/3}$ .) Next,

for all $\vec v\in S_j$ with $v_4-v_3\ge j^\upsilon $ (if j is large enough).

Let now $\vec v\in S_j$ with $v_2-v_1\ge j^\upsilon $ . Then applying directly Lemma 3.3 with $\ell _0=4$ , a similar reasoning gives $|\int _{\Omega _{*}^\varphi }\prod _{\ell =1}^4\xi _{v_\ell } \, da|\le C e_{[Cj^{5/3}]}$ .

Finally, since $\# S_j\le \# \mathbb {I}_j^4 \le j^{8/3}$ and $e_j\le j^{-d_1 \kappa _0+1}$ with $d_1 \kappa _0\ge 3>9/5$ , the bound in equation (4.20) gives C such that

(4.23) $$ \begin{align} E(w_j^4) \le C (j^{8/3}e_{[Cj^{5/3}]}+ j^{4/3 + 2\upsilon}) \le C j^{4/3+2\upsilon} \quad\text{for all } j\geq 1. \end{align} $$

(For the purposes of the present lemma, a version of equation (4.23) with $Cj^{5/3}$ in the right-hand side would suffice. The stronger statement is needed for equation (4.31).)

We next bound $|E (w_j^2 w_k^2) - E (w_j^2) E(w_k^2)|$ for $k\ge j+1$ . If $k = j + 1$ , by Cauchy’s inequality and equation (4.23),

$$ \begin{align*} E (w_j^2 w_{j+1}^2) \leq \sqrt{ E (w_j^4) E (w_{j+1}^4)} \leq C j^{5/3}. \end{align*} $$

By equation (3.8), we have $ E (w_{j}^2) E( w_{j + 1}^2) \leq C j^{4/3}$ . Hence,

(4.24) $$ \begin{align} |E (w_j^2 w_{j + 1}^2) - E (w_j^2) E( w_{j + 1}^2)| \leq C j^{5/3 }. \end{align} $$

Assume now that $k \geq j+2$ . By construction, $y_j$ is constant on elements of $\mathcal {P}_v$ if ${ v \geq r_{j_1}=j_1 + [j_1^\delta ]} $ , where $j_1$ is the largest number in $\mathbb {I}_j$ . Let

$$ \begin{align*} k_0=k_0(k):=\operatorname\min \mathbb{I}_{k} \ge \frac{k^{5/3}}{C}. \end{align*} $$

Then, for large enough j, using that $x\mapsto x-x^{1/4}$ is increasing for large x, we find

$$ \begin{align*} k_0 - k_0^{1/4} \geq j_1 + \#\mathbb{I}_{j+1} - (j_1 + \#\mathbb{I}_{j+1})^{1/4} \geq j_1 + 2 j_1^{2/3} - 2 j_1^{1/4} \geq j_1 + j_1^{1/4}. \end{align*} $$

Since $k\ge j+2$ and $\delta < \tfrac 14$ , we have that $y_j$ is constant on elements of $\mathcal {P}_v$ for

$$ \begin{align*} v=v(k_0)=[k_0-k_0^{1/4}]. \end{align*} $$

Lemma 2.3 gives $|x_{v (k_0)}(\omega )| \geq \unicode{x3bb} _0^{-{k_0}^{1/4}}$ if $\omega \in \mathcal {P}_{v(k_0)}$ . Thus, there exists a refinement $\mathcal {Q}_{v(k_0)}$ of $\mathcal {P}_{v(k_0)}$ such that

Therefore, for large enough k, the local bound in equation (3.9) in Proposition 3.2 gives for all $\omega \in \mathcal {Q}_{v}$ with non-empty intersection with $\Omega _{*}^\varphi $ that

$$ \begin{align*} \bigg| \frac{1}{|\omega|} \int_\omega w_k^2 \, dm -\# \mathbb{I}_k \bigg| \leq C , \end{align*} $$

since $n=\# \mathbb {I}_k=[k^{2/3}]\le \eta k_0/2$ . As in equation (3.10), we write $\mathcal {Q}_{*,v}$ for the set of $\omega \in \mathcal {Q}_{v}$ with non-empty intersection with $\Omega _{*}^\varphi $ , and $\Omega _{*,v}^{\mathcal {Q}} = \cup \mathcal {Q}_{*,v}$ . Thus, using that $y_j$ is constant on each $\omega \in \mathcal {Q}_v$ (since $\mathcal {Q}_{v}$ refines $\mathcal {P}_{v}$ ),

$$ \begin{align*} \int_{\Omega_{*,v}^{\mathcal {Q}}} (y_j^2 w_k^2) \, d m &= \sum_{\omega \in \mathcal{Q}_{v}^*} |\omega| \cdot y_j^2|_\omega \cdot \frac{1}{|\omega|} \int_\omega w_k^2\, dm\\ & \in \bigg[ \int_{\Omega_{*,v}^{\mathcal {Q}}} y_j^2 \, dm (\# \mathbb{I}_k - C), \int_{\Omega_{*,v}^{\mathcal {Q}}} y_j^2 \, dm (\# \mathbb{I}_k + C) \bigg]. \end{align*} $$

Recall that $j\le k-2$ . Since $|y_j^2| \le Cj^{4/3}\le Ck^{4/3}$ , we get

$$ \begin{align*} \frac 1 {m(\Omega_{*,v}^{\mathcal {Q}})}\int_{\Omega_{*,v}^{\mathcal {Q}}} y_j^2 \, d m = E(y_j^2) + O(1), \end{align*} $$

by equation (3.11) applied to $\Psi _k=y_j^2$ , and since $|y_j^2w_k^2| \le C k^{8/3}$ , we have

$$ \begin{align*} \frac 1 {m(\Omega_{*,v}^{\mathcal {Q}})}\int_{\Omega_{*,v}^{\mathcal {Q}}} (y_j^2 w_k^2) \, d m = E(y_j^2 w_k^2) + O(1), \end{align*} $$

by equation (3.11) applied to $\Psi _k=(y_j^2 w_k^2)$ . That is,

$$ \begin{align*} |E (y_j^2 w_k^2) - \#\mathbb{I}_k E (y_j^2)| \leq C (E (y_j^2) + 1). \end{align*} $$

Next, the global estimate in equation (3.8) in Proposition 3.2 gives $|E (y_j^2) E (w_k^2) - \# \mathbb {I}_k E (y_j^2)| \leq C E (y_j^2)$ . Therefore, (the expression $\#\mathbb {I}_j=j^b=j^{2/3}$ in the right-hand side already leads to $\gamma>2/5$ . A block size $\#\mathbb {I}_j=j^b$ replaces $3/5$ in equation (4.4) by $1/(1+b)$ , so that the first constraint becomes $N^\gamma> N^{b/(1+b)}$ , see equation (4.5). Our bounds in equations (4.25) and (4.26) (with Gál–Koksma’s strong law of large numbers, Theorem 4.3, and $M(N)\sim N^{1/(1+b)}$ ) give ${N^{\gamma }> N^{(b+2)/(4(b+1))}}$ . Hence, $b=2/3$ is the optimum. In the independent and identically distributed case, a block size $j^{1/2}$ gives $\gamma>1/3$ [PS, p. 25], see also the beginning of [Sch, §6])

$$ \begin{align*} |E (y_j^2 w_k^2) - E (y_j^2 )E (w_k^2)| \leq C (2E( y_j^2) + 1) )\leq C \# \mathbb{I}_j. \end{align*} $$

Hence, for large enough j and all $k\ge j+2$ , since $\sup |w_j+y_j|\le C \# \mathbb {I}_{j}$ ,

(4.25)

(We used equation (4.2) to get $\sup |w_j - y_j|\le C \# \mathbb {I}_j \tilde{\unicode{x3bb}} _\mathrm{CE}^{-\theta j^{5\delta /3}}$ .)

Finally, we plug equations (4.25), (4.24) and (4.23) into equation (4.19), and get, since ${2\upsilon <1/3}$ ,

(4.26)

This proves equation (4.18).

4.3 Martingale differences $Y_j$ . Skorokhod’s representation theorem

As in Schnellmann’s adaptation of [PS, §§3.4 and 3.5] in [Sch, §6.3], let $\mathcal {L}_j$ be the $\sigma $ -algebra generated by $\{y_\ell \}_{1\le \ell \le j}$ , and set

(4.27) $$ \begin{align} u_j &= \sum_{k \geq 0} E (y_{j + k} \mid \mathcal{L}_{j - 1}) , \quad Y_j = y_j + u_{j + 1} - u_j , \quad j\ge 2. \end{align} $$

Then, $\{Y_j,\mathcal {L}_j\}$ is a martingale difference sequence. Using equation (4.3), we show that $\{Y_j\}$ inherits the law of large numbers established for $\{y_j\}$ in Lemma 4.2:

Lemma 4.4. For $m_{*}$ -a.e. $a \in \Omega _{*}^\varphi $ , there exists $C(a)$ such that

(4.28) $$ \begin{align} \bigg| N -\sum_{j=1}^{M(N)} Y_j^2 (a) \bigg| \leq C(a)N^{2\gamma} \quad \text{for all } N \geq 1, \end{align} $$

and

(4.29) $$ \begin{align} \bigg| \sum_{j = 1}^{M(N)} E (Y_j^2 \mid \mathcal{L}_{j - 1}) - Y_j^2 (a) \bigg| \leq C(a) N^{2 \gamma} \quad \text{for all } N \geq 1. \end{align} $$

Proof. Recalling the $\sigma $ -algebra $\mathcal {F}_{r_i}$ generated by the intervals in $\mathcal {P}_{r_i}$ , we have ${\mathcal {L}_{\ell -1} \subset \mathcal {F}_{r_{i(\ell )}}}$ , where $i(\ell ) = \operatorname \max \{i \in \mathbb {I}_{\ell - 1} \} \leq C \ell ^{5/3}$ by equation (4.4). Then,

$$ \begin{align*} u_\ell = \sum_{j \geq 1} E ( E(\xi_{i(\ell) + j} \mid \mathcal{F}_{r_{i(\ell)}}) \mid \mathcal{L}_{\ell - 1}). \end{align*} $$

Since $\sum _{j=1}^\infty e_j < \infty $ , the bound in equation (4.3) in Lemma 4.1 gives

(4.30) $$ \begin{align} |u_\ell (a)| \leq \sum_{j \geq 1} C \operatorname\min \{1, e_{ [ \eta (j - 2 i(\ell)^\delta))]} \} \leq \frac {2 C}{ \eta} i(\ell)^\delta \leq C \ell^{5 \delta / 3}. \end{align} $$

Put $v_j = u_j - u_{j + 1}$ , so that $Y_j^2 = y_j^2 -2 y_j v_j + v_j^2$ .

We claim that equation (4.28) follows if for a.e. $a\in \Omega ^\varphi _{*}$ , there exists C such that $\sum _{j = 1}^{M(N)} v_j^2 \leq C N^{4 \gamma -1}$ . Indeed, since $\gamma <1/2$ , Lemma 4.2 and Cauchy’s inequality then give (using $\sum _{j = 1}^{M(N)} y_j^2\le C N$ )

$$ \begin{align*} \bigg| N - \sum_{j = 1}^{M(N)} Y_j^2 \bigg| & = \bigg| N - \sum_{j = 1}^{M(N)} (y_j^2 -2 y_j v_j + v_j^2) \bigg| \\ & \leq \bigg| N - \sum_{j = 1}^{M(N)} y_j^2 \bigg|+ \sum_{j=1}^{M(N)} v_j^2 + 2 \sqrt{\sum_{j = 1}^{M(N)} y_j^2 \sum_{j = 1}^{M(N)} v_j^2 } \\ & \leq C(a) N^{2 \gamma} + C N^{2 \gamma} + C \sqrt{N N^{4 \gamma-1}} \leq C(a) N^{2 \gamma}. \end{align*} $$

However, since we have $ v_j^2 \leq C j^{10\delta /3}$ (by equation (4.30)), we find, using $\delta <2(\gamma -2/5)$ ,

$$ \begin{align*} \sum_{j = 1}^{M(N)} v_j^2 \leq C M^{1 + 10 \delta / 3} \leq N^{3 / 5 + 2 \delta} \leq C N^{4 \gamma-1}. \end{align*} $$

It remains to prove equation (4.29). Set $R_j = Y_j^2 - E (Y_j^2 \mid \mathcal {L}_{j - 1})$ and observe that $\{R_j, \mathcal {L}_j\}$ is a martingale difference sequence. By Minkowski’s inequality

$$ \begin{align*} E (R_j^2) \leq \bigg(\sqrt{E (Y_j^4)} + \sqrt{E (E (Y_j^2 \mid \mathcal{L}_{j-1})^2)} \bigg)^2 \leq \bigg( 2 \sqrt{E ( Y_j^4)} \bigg)^2 = 4 E (Y_j^4). \end{align*} $$

Since $Y_j = y_j - v_j$ , we have, again by Minkowski’s inequality,

$$ \begin{align*} E (R_j^2) &\leq 4 E (Y_j^4) \leq 4 ( (E( y_j^4))^{\frac{1}{4}} + (E ( v_j^4))^{\frac{1}{4}} )^4 \leq C (E (y_j^4) + E (v_j^4)) \\ &\leq C (E (w_j^4 )+ E (|w_j^4 - y_j^4|) + E(v_j^4)). \end{align*} $$

Since $w_j^4 - y_j^4 = (w_j^2 + y_j^2) (w_j + y_j) (w_j - y_j)$ , we get from equation (4.2) that $E (|w_j^4 - y_j^4|)$ is uniformly bounded. By equation (4.30), we have $|u_j| \leq C j^{5\delta /3}$ . Hence, $|v_j| \leq |u_j| + |u_{j-1}| \leq C j^{5\delta /3}$ and $E (v_j^4) \leq C j^{20 \delta /3} \leq C j^{4/3}$ , since $\delta <1/5$ . For arbitrary $\iota>0$ , the bound in equation (4.23) gives C such that $E (w_j^4) \leq C j^{4/3 + \iota }$ . Thus,

(4.31) $$ \begin{align} \sum_{j \geq 1} \frac{E (R_j^2) }{j^{7/3 + \iota}}< \infty , \end{align} $$

and a martingale result (see [Ch]) implies that $\sum _{j\ge 1}R_j/j^{7/6+\iota }$ converges almost surely. For $m_{*}$ -a.e. $a \in \Omega _{*}^\varphi $ , Kronecker’s lemma gives $C(a)$ with

$$ \begin{align*} \sum_{j = 1}^{M(N)} R_j \leq C(a) M^{7/6 + \iota} \leq C(a) N^{21/30 + \iota} , \end{align*} $$

using equation (4.4) in the last inequality. Since $21 / 30 < 2 \gamma $ , this establishes equation (4.29).

We shall apply the following embedding result. (See [HH, Theorem A.1].)

Theorem 4.5. (Skorokhod’s representation theorem)

For any zero-mean square-integrable martingale $\{ \sum _{k=1}^j Y_k,\ \mathcal {L}_j \mid j \geq 1\} $ , there exist a probability space supporting a (standard) Brownian motion W, and non-negative variables $\{T_k, \, \, k \geq 1\}$ , such that $\{ \sum _{k=1}^j Y_k \}_{j \geq 1}$ and $\{ W(\sum _{k=1}^j T_k) \}_{j \geq 1}$ have the same distribution, and, in addition, letting $\mathcal {G}_0$ be the trivial $\sigma $ -algebra (the empty set and the entire space), and $\mathcal {G}_j$ , for $j\ge 1$ , be the $\sigma $ -algebra generated by

$$ \begin{align*} \{ W(t) \mid 0 \leq t \leq \tau_j\} \quad\mbox{where } \tau_j:=\sum_{k=1}^j T_k, \end{align*} $$

then $\tau _j$ is $\mathcal {G}_j$ -measurable, while $E (T_1 \mid \mathcal {G}_{0}) = E (W(T_1) ^2 \mid \mathcal {G}_{0})$ , and

$$ \begin{align*} E (T_j \mid \mathcal{G}_{j-1}) = E ( (W(\tau_j)-W(\tau_{j-1}) )^2 \mid \mathcal{G}_{j-1} ) \quad \text{for all } j\ge 2 , \mbox{ almost surely.} \end{align*} $$

By the last claim of Theorem 4.5 and properties of Brownian motion,

(4.32) $$ \begin{align} E (T_j \mid \mathcal{G}_{j - 1}) = E (W(T_j)^2 \mid \mathcal{G}_{j - 1}) \quad \text{for all } j\ge 1, \end{align} $$

almost surely. (Indeed, letting $W_1$ be an independent copy of W, we have ${W(\tau _j) = W_1 (\tau _{j-1} + T_j) = W_1 (\tau _{j-1}) + W (T_j)}$ in distribution, so that $W(\tau _j) - W(\tau _{j-1}) = W(T_j)$ in distribution.)

We need one last lemma. Recall that $\gamma \in (2/5,1/2)$ is fixed.

Lemma 4.6. (Strong law of large numbers for the sequence $T_j$ )

For $m_{*}$ -a.e. $a\in \Omega _{*}^\varphi $ , there exists $C(a)$ such that

(4.33) $$ \begin{align} \bigg| N - \sum_{j = 1}^{M(N)} T_j \bigg| \leq C(a) N^{2 \gamma} \quad \text{for all } N \geq 1. \end{align} $$

Proof. To start, apply Theorem 4.5 to the martingale difference sequence $Y_j$ from equation (4.27), with $\mathcal {L}_j$ generated by $\{y_\ell \}_{1\le \ell \le j}$ . Let $\tilde {Y}_j = W(\tau _j) - W(\tau _{j-1})$ , so that ${W (\tau _j) = \sum _{k=1}^j \tilde {Y}_{k}}$ and $\tilde {Y}_j = W(T_j)$ . By equation (4.32), we have, almost surely,

$$ \begin{align*} N - \sum_{j = 1}^{M(N)} T_j &= \bigg [N - \sum_{j = 1}^M \tilde{Y}_j^2 \bigg] + \sum_{j = 1}^M [\tilde{Y_j}^2 - E(\tilde{Y}_j^2 \mid\mathcal{G}_{j-1})] \\ &\quad + \sum_{j = 1}^M [ E(T_j \mid \mathcal{G}_{j-1}) - T_j ] \quad\text{for all } N\ge 1. \end{align*} $$

Then, since $Y_j$ and $\tilde {Y}_j$ have the same distribution, the bound in equation (4.28) in Lemma 4.4 gives $C(a)$ such that, for all $N\ge 1$ , the first sum in the right-hand side above is not larger than $C(a) N^{2\gamma }$ .

For the second sum in the right-hand side above, we use equation (4.29). Since conditional expectations can be expressed in terms of distributions, equation (4.29) is also valid with $Y_j$ replaced by $\tilde {Y}_j$ . Thus, the second sum in the right-hand side is also bounded by $C(a) N^{2\gamma }$ for all $N \geq 1$ .

Finally, let $R_j = E (T_j \mid \mathcal {G}_{j-1}) - T_j$ . Then, $\{R_j, \mathcal {G}_j\}$ is a martingale difference sequence by equation (4.32). As in the proof of equation (4.29), we can estimate $ E( R_j^2) \leq 4 E (W(T_j)^4)$ , and thus there exists $C(a)$ such that, for all $N\ge 1$ , we have $\sum _{j=1}^{M(N)} R_j \leq C N^{21/30 + \iota } \leq C(a) N^{2 \gamma }$ almost surely.

Proof of Theorem 1.1

Just like Schnellmann, we follow the proof of [PS, Lemma 3.5.3], replacing their $1/2-\alpha /2+\gamma $ by $\gamma $ , and replacing Lemma 3.5.1 there by our Lemma 4.6. We then obtain that, almost surely,

$$ \begin{align*} \bigg| \sum_{j = 1}^{M(N)} Y_j - W (N) \bigg| = O (N^\gamma). \end{align*} $$

Then, using equations (4.30) and (4.4), we find

(4.34) $$ \begin{align} \bigg| \sum_{j=1}^{M(N)} y_j - Y_j \bigg| = \bigg| \sum_{j=1}^{M(N)} (u_{j+1} - u_j) \bigg| = |u_{M(N)+1} - u_1| \le C N^{\delta}. \end{align} $$

Since $\delta <2/5$ , and recalling equation (4.5), this establishes Theorem 1.1.