# A parameter ASIP for the quadratic family

Magnus Aspenberg(1), Viviane Baladi(2),(3), and Tomas Persson(1) (1) Centre
for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden  (2)
Institute for Theoretical Studies, ETH, 8092 Zürich, Switzerland  (3) Sorbonne
Université and Université Paris Cité, CNRS, Laboratoire de Probabilités,
Statistique et Modélisation, F-75005 Paris, France
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

Consider the quadratic family $T_{a}(x)=ax(1-x)$, 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 transversal 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 $\sigma_{a_{*}}(\varphi)\neq 0$, there exists
$\epsilon_{\varphi}>0$ such that, for normalised 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 [BBS].

###### 2010 Mathematics Subject Classification:

37A10, 37A50, 37E05, 60F17

MA and TP thank the Institute for Theoretical Studies, ETH Zürich, for its
hospitality in 2022. VB’s research is supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 787304). VB is grateful to the K. and A.
Wallenberg Foundation, for an invitation to Lund University in 2021, and to J.
Sedro for the reference [Mo]. VB and TP thank Q. Berger and F. Pène for useful
conversations and acknowledge the hospitality of CIRM (Luminy) and the Lorentz
Center (Leiden), where part of this work was carried out in 2022.

###### Contents

  1. 1 Introduction
    1. 1.1 Background and Motivation
    2. 1.2 Statement of the ASIP (Theorem 1.1)
    3. 1.3 Structure of the Text
  2. 2 Bounds for the Quadratic Family. The Cantor Set $\Omega_{*}(a_{*})$
    1. 2.1 Basic Properties
    2. 2.2 A Polynomial Benedicks–Carleson Construction ($\Omega_{*}(a_{*})$, $\mathcal{P}_{n}$)
    3. 2.3 A Hölder Local Distortion Estimate
    4. 2.4 Uniform Decorrelation and Hölder Response
    5. 2.5 Hölder Regularity of the Variance $\sigma_{a}(\varphi)$
  3. 3 Switching Locally from the Parameter to the Phase Space
    1. 3.1 Proof of Proposition 3.2
  4. 4 Proof of Theorem 1.1 via Skorokhod’s Representation Theorem
    1. 4.1 Blocks $\mathbb{I}_{M}$. Approximations $\chi_{i}$ and $y_{j}$
    2. 4.2 Law of Large Numbers for $y_{j}^{2}$
    3. 4.3 Martingale Differences $Y_{j}$ — Skorokhod’s Theorem

##  1\. Introduction

###  1.1. Background and Motivation

Let $(\Omega_{*},m_{*},\mathcal{F}_{*})$ be a probability space. We say that a
sequence of measurable functions $\xi_{i}\colon\Omega_{*}\to{\mathbb{R}}$,
$i\geq 1$ 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 (centered) one-dimensional
Brownian motion $W$ and a sequence of measurable functions
$\eta_{i}\colon\Omega_{W}\to{\mathbb{R}}$, $i\geq 1$, such that

  1. i)

The random variables $\\{\xi_{i}\\}_{i\geq 1}$ and $\\{\eta_{i}\\}_{i\geq 1}$
have the same111By definition of the distribution of discrete-time real-valued
stochastic processes, this means that for any $n\geq 1$ and any
$\\{y_{i}\in{\mathbb{R}}\mid 1\leq i\leq n\\}$, the joint probability that
$\xi_{i}\leq y_{i}$ for all $1\leq i\leq n$ coincides with the joint
probability that $\eta_{i}\leq y_{i}$ for all $1\leq i\leq n$. distribution.

  2. ii)

Almost surely, $\bigl{|}W(n)-\sum_{i=1}^{n}\eta_{i}\bigr{|}=O(n^{\gamma})$ 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 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

$\limsup_{n\to\infty}\frac{1}{\sqrt{2n\log\log
n}}\sum_{i=1}^{n}\xi_{i}(a)=\sigma\,,\qquad\text{for $m_{*}$-almost every
}a\in\Omega_{*}\,,$

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

$\lim_{n\to\infty}m_{*}\biggl{(}\biggl{\\{}\,a\in\Omega_{*}\mid\frac{1}{\sigma\sqrt{n}}\sum_{i=1}^{n}\xi_{i}(a)\leq
y\,\biggr{\\}}\biggr{)}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{y}e^{-s^{2}/2}\,ds\,,\
\forall y\in{\mathbb{R}}\,.$

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

$T_{a}(x)=ax(1-x)\,,\quad x\in I\,,\,\,a\in(2,4]\,.$

Denote by $c=1/2$ the critical point of $T_{a}$ and set
$c_{j}(a)=T_{a}^{j}(c)$ for $j\geq 1$. We say that $T_{a}$ is mixing if it is
topologically mixing on the set

$K(a):=[c_{2}(a),c_{1}(a)]\,.$

It will be convenient below to restrict to mixing maps $T_{a}$. Tiozzo
recently showed [Ti, Cor 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 (and thus topologically mixing on $K(a)$ since the measure of
maximal entropy has full support there) if its topological entropy is greater
than $\log(2)/2$. Since the topological entropy of $T_{4}$ is equal to $\log
2$ and the topological entropy of $T_{a}$ is nondecreasing and continuous (in
fact Hölder continuous [Gu]) in $a$, there exists $a_{\mathrm{mix}}<4$ such
that $T_{a}$ is topologically mixing on $K(a)$ for all
$a\in(a_{\mathrm{mix}},4]$.

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 $a\in(a_{\mathrm{mix}},4]$ is
CE, then $T_{a}$ admits a unique absolutely continuous invariant probability
measure (acim) $\mu_{a}=h_{a}dm$. This measure is mixing, with support $K(a)$.
Our goal is to find a positive Lebesgue measure set
$\Omega_{*}\subset(a_{\mathrm{mix}},4]$ of (mixing) CE parameters with a
Lebesgue density point $a_{*}\in\Omega_{*}$ such that for any Hölder
continuous function $\varphi\colon I\to{\mathbb{R}}$ with
$\sigma_{a_{*}}(\varphi)\neq 0$ (see (1.2)), there exists
$\epsilon_{\varphi}>0$ such that the ASIP holds for $m_{*}$ the normalised
Lebesgue measure on
$\Omega_{*}\cap[a_{*}-\epsilon_{\varphi},a_{*}+\epsilon_{\varphi}]$ and

$\xi_{j}(a):=\varphi_{a}(c_{j+1}(a))\,,\qquad j\geq 0\,,\quad
a\in\Omega_{*}\cap[a_{*}-\epsilon_{\varphi},a_{*}+\epsilon_{\varphi}]\,,$

where $\varphi_{a}$ is a suitable normalisation of $\varphi$ (see (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–Smania [DLS] in the setting of piecewise expanding maps, in order
to study linear and fractional response. (This method requires a functional
central limit theorem, see [DLS, Lemma 5.1].)

Melbourne and Nicol showed [MN] 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, Section 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–Moreira [AM2], showed that222Benedicks and Carleson
established typicality in [BC1] for the Cantor set of CE parameters considered
there. 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)
$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\varphi(c_{i}(a))=\int_{0}^{1}\varphi\,d\mu_{a}\,,\qquad\forall\varphi\in
C^{0}\,.$

For Hölder continuous $\varphi\colon I\to{\mathbb{R}}$ and a topological
mixing CE parameter $a$, define $\sigma_{a}(\varphi)\geq 0$ by

(1.2) $\displaystyle(\sigma_{a}(\varphi))^{2}$
$\displaystyle:=\int_{0}^{1}\biggl{(}\varphi-\int\varphi
d\mu_{a}\biggr{)}^{2}\,d\mu_{a}$ (1.3)
$\displaystyle\qquad+2\sum_{i>0}\int_{0}^{1}\biggl{(}\varphi-\int\varphi\,d\mu_{a}\biggr{)}\biggl{(}\varphi-\int\varphi\,d\mu_{a}\biggr{)}\circ
T^{i}_{a}\,d\mu_{a}\,,$

where the sum (1.3) is finite because topological mixing (i.e., the fact that
the map is nonrenormalisable) implies [KN] exponential mixing for the acim and
Hölder continuous observables.

In a work in progress, Gao and Shen [GS2] show that, for Lebesgue almost every
$a$ in the set of transversal, 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 $\varphi$, i.e.,

$\limsup_{n\to\infty}\frac{1}{\sqrt{2n\log\log
n}}\sum_{i=1}^{n}\biggl{(}\varphi(T_{a}^{i}(c))-\int\varphi\,d\mu_{a}\biggr{)}=\sigma_{a}(\varphi)\,.$

###  1.2. Statement of the ASIP (Theorem 1.1)

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

$x_{j}(a)=c_{j+1}(a)=T^{j+1}_{a}(c)\,,\qquad
T_{a}^{\prime}(x)=\partial_{x}T_{a}(x)\,,\qquad
x_{j}^{\prime}(a)=\partial_{a}x_{j}(a)\,.$

The family $T_{a}$ is _transversal_ at ${a_{*}}$ if (see [Ts]) there exists
$C\geq 1$ such that

(1.4)
$\frac{1}{C}\leq\biggl{|}\frac{x_{j}^{\prime}(a_{*})}{(T_{a_{*}}^{j})^{\prime}(c_{1}(a_{*}))}\biggr{|}\leq
C\,,\qquad\forall j\geq 1\,.$

It is well known ([ALM], [Le], [AM1]) that there exists a full Lebesgue
measure333Since $T_{a}$ is a nondegenerate analytic family, in fact all but
countably many CE parameters are transversal. set $\Omega_{T}$ of CE
parameters such that $T_{a}$ is transversal at all $a_{*}\in\Omega_{T}$. We
refer to [Ts, ($NV_{t}$)] for an equivalent condition expressed in terms of
the postcritical orbit.

The map $T_{a}$ is $(H_{a},\kappa_{a})$-_polynomially recurrent,_ for
$\kappa_{a}\geq 1$ and $H_{a}\geq 1$, if

(1.5)
$|x_{j-1}(a)-c|=|T^{j}_{a}(c)-c|\geq\frac{1}{j^{\kappa_{a}}}\,,\qquad\forall
j\geq H_{a}\,.$

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. If $\inf_{j\geq
1}|T_{a}^{j}(c)-c|>0$ then $a$ is called a _Misiurewicz parameter._ The set of
Misiurewicz parameters $a$ is uncountable (it has full Hausdorff dimension
[Za, Thm. 1.4] but zero Lebesgue measure).

Finally, we introduce the normalisation $\varphi_{a}$: Let $\varphi$ be
bounded such that $\sigma_{a}(\varphi)\neq 0$ for a mixing CE parameter $a$.
Then the function

(1.6)
$\varphi_{a}(x):=\frac{1}{\sigma_{a}(\varphi)}\biggl{(}\varphi(x)-\int_{0}^{1}\varphi\,d\mu_{a}\biggr{)}\,$

is well defined and satisfies

(1.7)
$\sigma_{a}(\varphi_{a})=1\,\qquad\text{and}\quad\int\varphi_{a}\,d\mu_{a}=0\,.$

######  Theorem 1.1 (Main Theorem: ASIP).

For any transversal 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
$\sigma_{a_{*}}(\varphi)\neq 0$, there exists444The choice of
$\epsilon_{\varphi}$ ensures in particular that $\sigma_{a}(\varphi)\neq 0$ if
$\sigma_{a_{*}}(\varphi)\neq 0$. $\epsilon_{\varphi}>0$ such that the
functions

(1.8) $\xi_{n}(a):=\varphi_{a}(x_{n}(a))=\varphi_{a}(T_{a}^{n+1}(c))\,,\qquad
n\geq 1\,,$

satisfy the ASIP for normalised Lebesgue measure $m_{*}$ on
$\Omega_{*}\cap[a_{*}-\epsilon_{\varphi},a_{*}+\epsilon_{\varphi}]$ and all
error exponents $\gamma>2/5$.

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 $I$ (see e.g. Footnote
27). It is possible (but a bit cumbersome) to handle (a one-sided
neighbourhood of) this value by a change of coordinates as in [Ts, Lemma 2.1].

We expect that the methods555For example, [A, Lemma 8.1] would replace [DMS,
Lemma V.6.5] in the proof of Proposition 2.2. of this paper can be extended to
the case when the “root” $a_{*}$ is transversal, mixing, but only
Collet–Eckmann and polynomially recurrent (for large enough $\kappa_{0}>1$),
instead of Misiurewicz. We restrict here to Misiurewicz parameters $a_{*}$,
for the sake of simplicity, and, more importantly, since 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 exists $\beta>1$ such that

(1.9) $\lim_{\epsilon\to
0}\frac{m([a_{*}-\epsilon,a_{*}+\epsilon]\setminus\widetilde{\Omega})}{\epsilon^{\beta}}=0\,.$

(See [BS2, (5), Prop. F], note that [BS2, Lemma E] even uses $\beta<2$ close
to $2$.) Property (1.9) is known for all $\beta<2$ for the set
$\widetilde{\Omega}$ studied666Beware 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.org/2203.07942, is flawed.
by Tsujii [Ts]. Although it is not stated in the literature, the Cantor set
$\Omega_{BC}$ from the (exponential) Benedicks–Carleson construction at a
transversal Misiurewicz point $a_{*}$ should777See (2.18), noting that
$|\omega_{0}|=2\epsilon$, and taking $j\geq N_{0}$ for $N_{0}=O|\log\epsilon|$
(as is the case in the proof of Proposition 2.2). satisfy (1.9) for some
$\beta>1$. For our Cantor set $\Omega_{*}\subset\Omega_{BC}$, we expect that
for any $\kappa>1$, taking $\kappa_{0}$ large enough in Proposition 2.2 the
factor $\epsilon^{\beta}$ in (1.9) must be replaced by
$\epsilon|\log\epsilon|^{-\kappa}$ (see (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 at $a_{*}$ 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 — as opposed to enlarging the set of root
parameters $a_{*}$. If necessary, one could in fact restrict to postcritically
finite (Milnor–Thurston) root parameters $a_{*}$.

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, Prop. 4.3,
Lemma4.5]. 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, (III),
Theorem 3.2, Lemma 4.1, proof of Lemmas 6.1–6.2]) 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 (1.5) in our parameter exclusion (Proposition 2.2),
which causes the “thinness” of $\Omega_{*}$, is needed to apply the results of
[BBS] in Sections 2.4 and 2.5 (Propositions 2.5 and 2.6 on uniform
decorrelation and fractional response, and its consequence, Lemma 2.8). 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 the proof
of [Sch, Lemma 6.1] with (4.11) below). In addition, we get an easy proof of
the local distortion estimate (2.31). If the results 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
$\Omega^{\varphi}_{BC}$ as a baseline instead of $\Omega_{*}$ (if necessary,
the Benedicks–Carleson technique could be replaced by ideas from Tsujii [Ts],
Avila–Moreira [AM1] or Gao–Shen [GS1]). Next, one could try to _ignore_ the
parameters with exceptionally small images in Lemma 2.3. For (2.31), see also
Footnote 22.

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, Props 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, e.g.).

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 nonuniformly
expanding interval maps.

Our paper carries out this plan and is organised as follows: After recalling
basic facts in Section 2.1, we adapt in Section 2.2 the Benedicks and Carleson
procedure to construct, in a neighbourhood of a transversal 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 $\Omega_{*}(a_{*})=\cap_{n}\Omega_{n}$ is a
positive Lebesgue measure set of parameters satisfying the Collet–Eckmann
property, polynomial returns and distortion control, with uniform constants.
(Our distortion bound (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 (Sections
4.1 and 4.2), then (Section 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, Section 3]). The usual
application of the approach of [PS, Chapter 7] in dynamics uses a strong
independence condition (see [PS, 7.1.2]) which we do not have (the $\xi_{i}$’s
are not iterations of a fixed map and there is no888See the example (see [Ka],
p. 646) discussed in [Sch]. Also, as pointed out by [Sch], it is not clear how
to apply the spectral techniques of [Go] in our setting. underlying invariant
measure). 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,
Section 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
$|\varphi(x)-\varphi(y)|\leq H_{\varpi}|x-y|^{\varpi}$ 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

$a\mapsto T^{\prime}_{a}(x)=\partial_{x}T_{a}(x)=a(1-2x)\,,\qquad
x\mapsto\partial_{a}T_{a}(x)=x(1-x)$

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

(2.1) $\sup_{x\in I}|T_{a}^{\prime}(x)|\leq\Lambda:=4\,,\qquad\forall
a\in(a_{\mathrm{mix}},4]\,.$

Each $T_{a}$ has two monotonicity intervals, with partition points $0$,
$c=1/2$, and $1$. The following easy lemma replaces999We 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]. [Sch, (30)]:

######  Lemma 2.1.

There exists $C<\infty$ such that, for any $a_{1},a_{2}\in(2,4]$, we have

(2.2) $|T_{a_{1}}^{n}(x)-T_{a_{2}}^{n}(x)|\leq
C\Lambda^{n}|a_{1}-a_{2}|\,,\qquad\forall x\in I\,,\quad\forall n\geq 1\,.$

###### Proof.

Clearly, $|T_{a_{1}}(x)-T_{a_{2}}(x)|\leq|a_{1}-a_{2}|$. For $n\geq 2$, using
the definition (2.1) of $\Lambda$, and setting
$C=\sum_{j=0}^{\infty}\Lambda^{-j}$, we get

$\displaystyle|T_{a_{1}}^{n}(x)$ $\displaystyle-T_{a_{2}}^{n}(x)|$
$\displaystyle\leq|T_{a_{1}}(T_{a_{1}}^{n-1}(x))-T_{a_{2}}(T_{a_{1}}^{n-1}(x))|+|T_{a_{2}}(T_{a_{1}}^{n-1}(x))-T_{a_{2}}(T_{a_{2}}^{n-1}(x))|$
$\displaystyle\leq|a_{1}-a_{2}|+\Lambda|T_{a_{1}}^{n-1}(x)-T_{a_{2}}^{n-1}(x)|$
$\displaystyle\leq|a_{1}-a_{2}|(1+\Lambda)+\Lambda|T_{a_{1}}^{n-2}(x)-T_{a_{2}}^{n-2}(x)|\leq\ldots$
$\displaystyle\leq|a_{1}-a_{2}|\sum_{j=0}^{n-1}\Lambda^{j}\leq
C\Lambda^{n}|a_{1}-a_{2}|\,.\qed$

###  2.2. A Polynomial Benedicks–Carleson Construction ($\Omega_{*}(a_{*})$,
$\mathcal{P}_{n}$)

For each $j\geq 0$, 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 (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 and Carleson Cantor
set101010See (2.17) for the construction of $\Omega_{BC}$.
$\Omega_{BC}=\Omega_{BC}(a_{*})$ (see [BC1, BC2]) associated to a transversal
Misiurewicz parameter $a_{*}$:

######  Proposition 2.2 (The Cantor set
$\Omega_{*}=\Omega_{*}(a_{*},\kappa_{0})$).

Let $a_{*}\in(a_{\mathrm{mix}},4]\cap\Omega_{T}$ be a Misiurewicz parameter.
There exist $\lambda_{CE}\in(1,\Lambda)$ and $C_{0}\in(0,1)$ such that, for
any $d_{1}\in(0,C_{0}\log\lambda_{CE}/4)$ and $d_{0}>0$, there exists
$\epsilon>0$ such that, for any $\kappa_{0}>1/d_{1}$, there exist $N_{0}\geq
1$ and a sequence $\mathcal{P}_{j}$ of finite sets of pairwise disjoint
subintervals of

$\omega_{0}:=[a_{*}-\epsilon,a_{*}+\epsilon]\cap(a_{\mathrm{mix}},4]\,,$

such that $\mathcal{P}_{1}=\mathcal{P}_{2}=\ldots=\mathcal{P}_{N_{0}}$, while
$\mathcal{P}_{j+1}\subset\mathcal{P}_{j}$ for $j\geq N_{0}$, and111111The
first bound of (2.4) implies that $a\mapsto x_{j}(a)=T_{a}^{j+1}(c)$ is
monotone on $\omega\in\mathcal{P}_{j}$.

(2.3) $\displaystyle\forall j\geq
1\,,\quad\forall\omega\in\mathcal{P}_{j}\,,\quad\forall
0\leq\ell<j\,,\quad\exists\omega^{\prime}\in\mathcal{P}_{\ell}\mbox{ such that
}\omega\subset\omega^{\prime}\,,$ (2.4)
$\displaystyle|x^{\prime}_{j}(a)|>0\,,\quad|T_{a}^{j+1}(c)-c|>0\,,\quad\forall
a\in\omega\,,\quad\forall\omega\in\mathcal{P}_{j}\,,\quad\forall j\geq 0\,,$

and there exists121212Note that (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)
$\displaystyle|(T_{a}^{n})^{\prime}(T_{a}(c))|\geq\lambda_{CE}^{n}\,,\qquad$
$\displaystyle\forall N_{0}\leq n\leq j\,,\quad\forall a\in\omega\,,$ (2.6)
$\displaystyle\frac{1}{C}\leq\biggl{|}\frac{x_{n}^{\prime}(a)}{(T_{a}^{n+1})^{\prime}(c)}\biggr{|}\leq
C\,,\qquad$ $\displaystyle\forall N_{0}\leq n\leq j\,,\quad\forall
a\in\omega\,,$ (2.7) $\displaystyle|\tilde{\omega}|\leq
C\lambda_{CE}^{-n}|x_{n}(\tilde{\omega})|\,,\qquad$ $\displaystyle\forall
N_{0}\leq n\leq j\,,\quad\forall\tilde{\omega}\subset\omega\,,$

and, moreover,

(2.8) $|T_{a}^{n+1}(c)-c|>n^{-\kappa_{0}}\,,\qquad\forall N_{0}\leq n\leq
j\,,\quad\forall a\in\omega\,.$

Finally, setting

$\Omega_{*}=\Omega_{*}(a_{*},\kappa_{0}):=\bigcap_{j\geq
N_{0}}\Omega_{j}\,,\qquad\text{with}\qquad\Omega_{j}:=\bigcup_{\omega\in\mathcal{P}_{j}}\omega\,,$

we have that $a_{*}\in\Omega_{*}$ is a Lebesgue density point of
$\Omega_{*}\subset\Omega_{T}$, with

(2.9) $|\Omega_{*}|\geq(1-d_{0}\cdot e_{j})|\Omega_{j-1}|\,,\qquad\forall
j\geq N_{0}\,,\quad\text{where}\quad
e_{j}:=\sum_{n=j}^{\infty}n^{-d_{1}\cdot\kappa_{0}}\,,$

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

(2.10) $\sum_{\begin{subarray}{c}\omega\in\mathcal{P}_{\ell}\\\
\omega\subset\omega^{\prime}\end{subarray}}|\omega\setminus(\omega\cap\Omega_{*})|\leq
d_{0}\cdot
e_{\ell-\ell^{\prime}}|\omega^{\prime}|\,,\quad\forall\omega^{\prime}\in\mathcal{P}_{\ell^{\prime}}\,,\,\forall\ell\geq\ell^{\prime}\geq
N_{0}\,.$

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

Clearly, (2.8) means that any $a\in\Omega_{*}$ is
$(N_{0},\kappa_{0})$-polynomially recurrent.

The bound (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 $d_{1}\cdot\kappa_{0}\geq 3$ to use (2.9) in the proof of
Proposition 3.2 (and $d_{1}\cdot\kappa_{0}>9/5$ for Lemma 4.2).

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

###### Proof of Proposition 2.2.

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

$I_{r}^{+}=[c+e^{-r-1},c+e^{-r})\,,\,\,I_{r}^{-}=(c-e^{-r},c-e^{-r-1}]\,,\,\,U_{r}=(c-e^{-r},c+e^{-r})\,,$

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 $c$. Let131313The constant $\alpha_{BC}$ is usually
called $\alpha$, but we shall need the letter $\alpha$ for another purpose in
(2.30). $\beta_{BC}>\alpha_{BC}>0$ where

$e^{-n\alpha_{BC}}\leq n^{-\kappa_{0}}\,,\,\,\forall n\geq N_{0}\,,$

for $N_{0}$ a large integer to be chosen later.

For $a\in(a_{\mathrm{mix}},4]$, $\nu\geq 1$, and $r\geq r_{0}$ 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

$|T^{j}_{a}(x)-T^{j+\nu}_{a}(c)|\leq e^{-j\beta_{BC}}\,,\qquad\forall 1\leq
j\leq p\,,\quad\forall x\in U_{r}\,.$

The first _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
$i\geq 2$, define inductively the $i$th _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

$T^{j}_{a}(c)\cap
U_{r_{0}}=\emptyset\,,\qquad\forall\nu_{i-1}(\omega)+p_{i-1}(\omega)+1\leq
j<\nu_{i}(\omega)\,,\,\,\forall a\in\omega\,,$

and, for $r(\omega)$ such that $x_{\nu_{i-1}-1}(\omega)\subset I_{r}$, set the
$i$th _binding time_ of $\omega$ to be

$p_{i}(\omega)=\operatorname{min}_{a\in\omega}p(r,a,\nu_{i-1}(\omega))\,.$

(Similarly, define inductively for $i\geq 2$ and $a$ such that
$T^{\nu_{i-1}}_{a}(c)\subset I_{r}$, 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 $i$th bound period of $\omega$, those
between $\nu_{i-1}(\omega)+p_{i-1}(\omega)+1$ and $\nu_{i}(\omega)-1$ form its
$i$th free period. Finally, if there exist $a\in\omega$ and
$j\geq\nu_{1}(\omega)$ 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.)

For small $\epsilon>0$ (to be chosen depending on $r_{0}$), the sequence
$\mathcal{P}_{j}$ can now be defined141414We refer throughout to [DMS, Section
V.6]. The original ideas and key estimates appeared previously in the work of
Benedicks and Carleson [BC1, BC2]. See Footnote 16. inductively: Start with
the single interval
$\mathcal{P}_{0}=\mathcal{P}_{1}=\ldots=\mathcal{P}_{N_{0}}=\\{\omega_{0}\\}$,
for $\epsilon$ small enough such that $\nu_{1}(\omega_{0})\geq N_{0}$ (note
that $\nu_{1}(\omega_{0})$ increases if $r_{0}$ increases or $\epsilon$
decreases). Let $W_{a_{*}}$ be a neighbourhood of $c$ disjoint from
$\\{T^{n}_{a_{*}}(c)\mid n\geq 1\\}$. By taking $r_{0}$ large, we require that
$x_{\nu_{1}(\omega_{0})-1}(\omega_{0})$ contains $W_{a_{*}}$ (see [DMS,
beginnings of Steps 2 and 3]). Note that there exists $r_{1}\geq r_{0}$ such
that $U_{r_{1}}\subset W_{a_{*}}$.

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 return151515That is, either $j$ is not a return, or it is
a return within the bound period. time of $\omega$ we include $\omega$ in
$\mathcal{P}^{\prime}_{j}$. 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:

$x_{j-1}(\omega)\setminus U_{r_{0}}\,,\qquad\\{x_{j-1}(\omega)\cap
I_{r,\ell}^{\pm}\mid r\geq r_{0}\,,\,1\leq\ell\leq r^{2}\\}\,.$

If $x_{j-1}(\omega)\setminus U_{r_{0}}\neq\emptyset$, 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 $x_{j-1}(\omega)\cap
I_{r,\ell}^{\pm}\neq\emptyset$, 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 $\\{\hat{\omega}_{r,\ell}\mid r\geq r_{0}-1\\}$ 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 $r\geq
r_{0}$, 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) $e^{r}\geq(j-1)^{\kappa_{0}}\,.$

Mapping the remaining intervals via the inverse of the diffeomorphism (see
[DMS, Prop. 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, Section
V.6], [Mo], which finally defines $\mathcal{P}_{j}$. For further use, we
denote these remaining intervals by

(2.12) $\omega_{r,\ell}=x_{j-1}^{-1}(\hat{\omega}_{r,\ell})\,.$

It is well known161616The original construction in [BC1, BC2] is for
$a_{*}=4$, see [Mo] for a self-contained account. It extends to transversal
Misiurewicz parameters: for CE parameters, the condition in [DMS, Theorem 6.1]
is equivalent to (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 (2.11) (used
to discard intervals) by the exponential condition

(2.13) $\omega\cap I_{r,\ell}^{\pm}\neq\emptyset\quad\mbox{and}\quad e^{r}\geq
e^{\alpha_{BC}(j-1)}\,,$

to construct sequences $\mathcal{P}_{j}^{{}^{\prime},BC}$ and
$\mathcal{P}_{j}^{BC}$, then there exists $\lambda_{CE}>1$ (called
$e^{\gamma}$ in [DMS, (V.6.4), Theorem V.6.2]) such that for any small enough
$\beta_{BC}>\alpha_{BC}>0$ there exist $N_{0}^{\prime}$ such that, if $r_{0}$
is large enough and $\epsilon>0$ small enough, then the $\mathcal{P}_{j}^{BC}$
satisfy (2.3)–(2.7) ((2.5) is called $(EX_{j})$ in [DMS, Section V.6]) for
some $C<\infty$, and the following condition (noted171717Strictly speaking,
the condition $(BA_{j})$ does not involve the factor $2$, and a condition
$(BA^{\prime}_{j})$ requiring that for each
$\omega\in\mathcal{P}_{j}^{{}^{\prime}BC}$ there exists $a\in\omega$ with
$|T_{a}^{n+1}(c)-c|>e^{-n\alpha_{BC}}$ for $N_{0}^{\prime}\leq n\leq j$ is
used in some lemmas. See [DMS, Section V.6, Step 5]. $(BA_{j})$ in the
literature) holds for all $j\geq N_{0}^{\prime}$

(2.14) $\displaystyle 2|T_{a}^{n+1}(c)-c|>e^{-n\alpha_{BC}}\,,\qquad\forall
N^{\prime}_{0}\leq n\leq j\,,\ \forall a\in\omega\,\
\forall\omega\in\mathcal{P}_{j}^{{}^{\prime},BC}\,.$

Since $\lambda_{CE}$ does not depend on $\alpha_{BC}$, $N_{0}$, or
$N_{0}^{\prime}$, we may assume that

$4\alpha_{BC}<\log\lambda_{CE}$

and we may replace $N_{0}$ by $\operatorname{max}\\{N_{0},N_{0}^{\prime}\\}$.

In particular [DMS, Prop. V.6.1, Lemma V.6.1 b), c)] give $\gamma_{0}>0$,
$\lambda_{CE}=e^{\gamma}\in(1,e^{\gamma_{0}})$, and $C_{0}>0$ (independent of
$r_{0}$ and $\epsilon$) such that, if $a\in\Omega_{n}$ and
$\nu_{\ell+1}(a)\leq n$, writing $p_{\ell}$, $\nu_{\ell}$ for $p_{\ell}(a)$,
$\nu_{\ell}(a)$, we have

(2.15)
$\begin{cases}|(T_{a}^{\nu_{\ell+1}-(\nu_{\ell}+p_{\ell}+1)})^{\prime}(T^{\nu_{\ell}+p_{\ell}+1}_{a}(c))|\geq
C_{0}e^{\gamma_{0}(\nu_{\ell+1}-(\nu_{\ell}+p_{\ell}))}\\\
|(T_{a}^{p_{\ell}+1})^{\prime}(T^{\nu_{\ell}}_{a}(c))|\geq\lambda_{CE}^{p_{\ell}/4}\,.\end{cases}$

To establish (2.5) (the bound below will also be used for (2.34)), one takes
$r_{0}$ such that

$r_{0}^{2}C_{0}^{2}\log\lambda_{CE}>|\log C_{0}|\,.$

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

(2.16)
$\biggl{|}\frac{x_{j}^{\prime}(a_{1})}{x_{j}^{\prime}(a_{2})}\biggr{|}\leq
C\,,\qquad\forall N_{0}\leq j\leq n\,,\quad\forall a_{1},a_{2}\in\omega\,,$

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 (2.6) follows from [DMS, Prop.
V.6.2 and Theorem V.6.2].

It is easy to check that (2.11) implies (2.8) (for returns during a bound
period, use that $\ell^{-\kappa_{0}}-e^{-\ell\beta_{BC}}\geq j^{-\kappa_{0}}$
for all $N_{0}\leq\ell\leq j-1$, up to increasing $N_{0}$ again). Our choice
of $N_{0}$ implies $\Omega_{j}\subset\Omega_{j}^{BC}$. Also, (2.6) with (2.4)
imply $\Omega_{*}\subset\Omega_{T}$. Since (2.7) is an immediate consequence
of (2.5)–(2.6), it only remains to establish that $a_{*}$ is a Lebesgue
density point in $\Omega_{*}$ (clearly, $a_{*}\in\Omega_{*}$) and that (2.9)
and (2.10) hold.

To show that $a_{*}$ is a Lebesgue density point of $\Omega_{*}$, we may
follow181818We 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]. [DMS,
Step 7 of the proof of Theorem V.6.1], replacing $Ce^{-iC_{0}}$ there by
$C^{\prime}i^{-\kappa_{0}}$.

To establish (2.9) and (2.10), let
$\Omega^{\prime}_{j}:=\bigcup_{\omega\in\mathcal{P}^{\prime}_{j}}\omega$,
recall $\Omega_{j}$, and define $\Omega_{j}^{BC}$ and
$\Omega_{j}^{{}^{\prime},BC}$ accordingly, setting

(2.17)
$\Omega_{BC}=\Omega_{BC}(a_{*},\alpha_{BC})=\cap_{j}\Omega^{BC}_{j}\,,\quad\mbox{so
that}\quad\Omega_{*}(a_{*})\subset\Omega_{BC}(a_{*})\,.$

For suitably small $\bar{\eta}>0$, and for $J_{0}\geq 1$ such that191919Since
$\bar{\eta}$ is independent of $\epsilon$, $r_{0}$, $N_{0}$, we may take
$N_{0}\geq J_{0}$. $\prod_{j=J_{0}}^{\infty}(1-e^{-\bar{\eta}j})>3/4$, the
parameter exclusion rule (2.13) gives $d^{\prime}_{0}>0$ (tending to zero with
$\epsilon$) such that ([DMS, Section V.6, Step 7], [Mo, §6])

(2.18)
$\displaystyle\begin{cases}|\omega\cap\Omega^{{}^{\prime},BC}_{j}|\geq(1-d^{\prime}_{0}e^{-j\bar{\eta}})|\omega|\,,&\forall\omega\in\mathcal{P}^{BC}_{j-1}\,,\,\,\forall
j\geq J_{0}\,,\\\
|\Omega^{BC}_{j}|\geq|\Omega^{{}^{\prime},BC}_{j}|-e^{-j\bar{\eta}}|\omega_{0}|\,,\,\,&\forall
j\geq J_{0}\,.\end{cases}$

The above implies
$|\Omega^{BC}_{j}|\geq(1-d^{\prime}_{0}e^{-\bar{\eta}j})|\Omega^{BC}_{j-1}|-e^{-\bar{\eta}j}|\omega_{0}|$
for $j\geq J_{0}$, and, exploiting that $|\omega_{0}|=|\Omega^{BC}_{n}|$ for
all $n\leq N_{0}$ with $N_{0}\geq J_{0}$, and using the definition of $J_{0}$,
also that

$|\Omega^{BC}_{j}|\geq\biggl{(}\prod_{n=J_{0}}^{j}(1-d^{\prime}_{0}e^{-\bar{\eta}n})-\sum_{n=J_{0}}^{j}e^{-\tilde{\eta}n}\biggr{)}|\omega_{0}|\geq\frac{1}{2}|\omega_{0}|\,,\qquad\forall
j\geq J_{0}\,.$

(By taking larger $J_{0}$, i.e. smaller $\epsilon$, we could replace $1/2$ by
a number close to $1$.) Thus, applying inductively

$|\Omega^{BC}_{j}|\geq\bigl{(}(1-d^{\prime}_{0}e^{-\bar{\eta}j})-2e^{-\bar{\eta}j}\bigr{)}|\Omega^{BC}_{j-1}|\,,\qquad\forall
j\geq J_{0}\,,$

we find $\tilde{\eta}>0$ such that for any $j\geq J_{0}$

(2.19)
$|\Omega_{BC}|\geq\prod_{n=j}^{\infty}(1-(d^{\prime}_{0}+2)e^{-\bar{\eta}n})|\Omega_{j-1}^{BC}|\geq(1-(\tilde{d}_{0}+2)e^{-\tilde{\eta}j})|\Omega_{j-1}^{BC}|\,.$

Recall that we fixed $d_{1}\in(0,\frac{C_{0}}{4}\log\lambda_{CE})$
(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 (2.11) for $\kappa_{0}>1/d_{1}$. We claim that for any
$d_{0}>0$, if $\epsilon$ is small enough,

(2.20)
$\displaystyle\begin{cases}|\omega\cap\Omega^{\prime}_{j}|\geq(1-d_{0}\cdot
j^{-d_{1}\kappa_{0}})|\omega|\,,&\forall\omega\in\mathcal{P}_{j-1}\,,\,\,\forall
j\geq J_{1}\,,\\\
|\Omega_{j}|\geq|\Omega^{\prime}_{j}|-e^{-j\bar{\eta}}|\omega_{0}|\,,&\forall
j\geq J_{1}\,.\end{cases}$

Before establishing this claim, we note that, mutatis mutandis, (2.20)
combined with the arguments leading to (2.19) implies (2.9), while the more
precise claim (2.10) follows from the refinement of (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–Prop. 6.5]).

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

Firstly, 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 (2.8), becomes

(2.21) $\frac{|\omega\setminus\bigcup_{r\geq\kappa_{0}\log
j}\,\omega_{r,\ell}|}{|\omega|}\geq 1-Cj^{-d_{1}\kappa_{0}}\,,\qquad\forall
j\geq N_{0}\,.$

To show this first claim, use that the constant $C_{0}\in(0,1)$ (introduced
above) is independent of $\kappa_{0}$ (because $\lambda_{CE}$ 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) $I_{r^{\prime},\ell^{\prime}}\subset
x_{\nu}(\omega)\,,\quad\mbox{for}\quad r_{0}\leq
r^{\prime}\leq\kappa_{0}\log\nu\leq\kappa_{0}\log j\,,$

satisfies $p\geq C_{0}r^{\prime}$. 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)
$|x_{j}(\omega)|\geq\lambda_{CE}^{p/4}\frac{e^{-r^{\prime}}}{(r^{\prime})^{2}}\geq\frac{e^{(-1+d_{1})r^{\prime}}}{(r^{\prime})^{2}}\geq\frac{1}{j^{\kappa_{0}(1-d_{1})}}\,,\qquad
j\geq N_{0}\,,$

where we used $d_{1}\leq\frac{C_{0}}{4}\log\lambda_{CE}$ in the second
inequality. We can thus replace the chain of inequalities after [DMS,
V.(6.20)] (using the distortion bound (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 (2.23) also holds
for $\tilde{\omega}$) by

$\frac{|\bigcup_{r\geq\kappa_{0}\log
j}\,\omega_{r,\ell}|}{|\omega|}\leq\frac{|\bigcup_{r\geq\kappa_{0}\log
j}\,\omega_{r,\ell}|}{|\tilde{\omega}|}\leq
C\frac{1}{j^{\kappa_{0}}}\frac{1}{|x_{j}(\tilde{\omega})|}\leq
Cj^{-d_{1}\cdot\kappa_{0}}\,.$

Secondly,202020We mention here a typo: [DMS, V.(6.24)] follows from [DMS,
V.(6.22)] (and not [DMS, V.(6.20)] as stated there). [DMS, Lemma V.6.6] (which
deals with $(FA_{j})$) uses (2.14) only via [DMS, Lemma V.6.3], while [DMS,
Lemma V.6.3] still holds (with the same proof) if we replace (2.14) by our
stronger assumption (2.8).

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

Lemma 2.3 below is the analogue of [Sch, (III)’]):

######  Lemma 2.3 (No Exceptionally Small Sets).

For any $\kappa_{1}>\kappa_{0}$ there exists $N_{1}\geq N_{0}$ such that
$|x_{j}(\omega)|>j^{-\kappa_{1}}$ for all $j\geq N_{1}$ 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 $j\geq N_{0}$, and any $\omega\in\mathcal{P}_{j}$, we have

(2.24) $|x_{j}(\omega)|\geq\frac{d_{2}e^{-r_{0}}(1-1/e)}{(\kappa_{0}\log
j)^{2}j^{\kappa_{0}}}\,,$

with $r_{0}$ as in the proof of Proposition 2.2. Indeed (2.24) implies that

$|x_{j}(\omega)|\geq\frac{d_{2}e^{-r_{0}}(1-e^{-1})}{\kappa_{0}^{2}}\frac{1}{j^{\kappa_{0}}(\log
j)^{2}}\,,\qquad\forall\omega\in\mathcal{P}_{j}\,,\forall j\geq N_{0}\,.$

Clearly, there exists $N_{1}(\kappa_{1})\geq N_{0}$ such that the right-hand
side is larger than $j^{-\kappa_{1}}$ for all $j\geq N_{1}$.

To establish (2.24), we shall use (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)
$|x_{j}(\omega)|\geq|I_{r,\ell}^{\pm}|=e^{-r}\frac{1-1/e}{r^{2}}>\frac{j^{-\kappa_{0}}(1-1/e)}{(\kappa_{0}\log
j)^{2}}\,.$

Otherwise, letting $j^{\prime}+1=\nu_{i^{\prime}}(\omega)\geq\nu_{1}(\omega)$
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
(2.25) implies

$|x_{j^{\prime}}(\omega)|>\frac{1-1/e}{(\kappa_{0}\log
j^{\prime})^{2}(j^{\prime})^{\kappa_{0}}}>\frac{1-1/e}{(\kappa_{0}\log
j)^{2}j^{\kappa_{0}}}\,.$

We shall combine the above bound with [DMS, Lemma V.6.3, Props V.6.1–6.2] 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 $i^{\prime}\leq i$ as defined above,

(2.26) $|x_{j}(\omega)|\geq
2^{i-i^{\prime}}|x_{j^{\prime}}(\omega)|>2^{i-i^{\prime}}\frac{1-1/e}{(\kappa_{0}\log
j)^{2}j^{\kappa_{0}}}\,.$

If $j+1$ is a return within the bound period of a previous free return
$j^{\prime\prime}+1$ of $\omega$, then using (2.25) for the bound period of an
essential return, respectively (2.26) for the bound period of a nonessential
return, and applying the first claim of [DMS, Lemma V.6.3], we find
$d_{2}\in(0,1)$ such that

(2.27) $|x_{j}(\omega)|\geq
d_{2}\lambda_{CE}^{j-j^{\prime\prime}}|x_{j^{\prime\prime}}(\omega)|>\frac{d_{2}(1-1/e)}{(\kappa_{0}\log
j)^{2}j^{\kappa_{0}}}\,.$

If $x_{j}(\omega)\cap U_{r_{0}}=\emptyset$ then [DMS, V.(6.2) in Prop. V.6.1
and Prop. V.6.2] and (2.25) give

(2.28) $|x_{j}(\omega)|\geq
d_{2}e^{-r_{0}}|x_{j^{\prime}}(\omega)|>\frac{d_{2}e^{-r_{0}}(1-1/e)}{(\kappa_{0}\log
j)^{2}j^{\kappa_{0}}}\,.$

We have shown (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 transversal Misiurewicz
parameter, fix $\kappa_{0}\geq 3/d_{1}(a_{*})$, and let
$\Omega_{*}=\Omega_{*}(a_{*},\kappa_{0})\subset\Omega_{BC}=\Omega_{BC}(a_{*})$
be the positive measure Cantor set constructed in Section 2.2 via families
$\mathcal{P}_{j}=\mathcal{P}_{j}(a_{*},\kappa_{0})$. The following212121For
[Sch, (30)], see (2.2). We do not need [Sch, (32)]. replaces [Sch, (33),
(31)]. The bound (2.31) is new.

######  Lemma 2.4 (Hölder Distortion Bounds).

There exists $C<\infty$ such that for all $n\geq N_{0}$ (with $N_{0}$ as in
Proposition 2.2) and any
$\omega\in\mathcal{P}_{n}=\mathcal{P}_{n}(a_{*},\kappa_{0})$

(2.29)
$\frac{1}{C}\leq\biggl{|}\frac{x_{n}^{\prime}(a)/x_{j}^{\prime}(a)}{(T_{a}^{n-j})^{\prime}(x_{j}(a))}\biggr{|}\leq
C\,,\qquad\forall 1\leq j\leq n\,,\quad\forall a\in\omega\,.$

In addition, there exist $C<\infty$ and $M_{0}>\kappa_{0}$ such that, for all
$n\geq N_{0}$, 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) $|x_{n}(\omega)|\leq n^{-M_{0}/(1-\alpha)}\,,$

we have

(2.31)
$\biggl{|}\frac{x_{n}^{\prime}(a_{1})}{x_{n}^{\prime}(a_{2})}\biggr{|}\leq
1+C|x_{n}([a_{1},a_{2}])|^{\alpha}\,,\qquad\forall a_{1},a_{2}\in\omega\,.$

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

###### Proof.

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

We first claim that222222Our 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 (2.31). there exist $C^{\prime}$ and $\kappa_{2}>0$
such that for any $n$

(2.32) $\sum_{i=0}^{j-1}|x_{i}(\omega)|\leq
C^{\prime}j^{\kappa_{0}+1+\kappa_{2}}|x_{j}(\omega)|\,,\quad\forall 1\leq
j\leq n\,,\,\forall\omega\subset\tilde{\omega}\in\mathcal{P}_{n}\,.$

To start, there is $C$ such that for any $0\leq i\leq j\leq n$, using (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) $\displaystyle\frac{|x_{i}(\omega)|}{|x_{j}(\omega)|}$
$\displaystyle=\frac{|x_{i}(\omega)|}{|X_{i,j}(x_{i}(\omega))|}=\frac{|x^{\prime}_{i}(a)|}{|x^{\prime}_{j}(a)|}\leq\frac{C}{|(T^{j-i}_{a})^{\prime}(T^{i+1}_{a}(c))|}\,.$

(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 $\nu_{\ell}(a)\leq j$, and put232323The condition $(FA)_{n}$ implicitly
used in Proposition 2.2 says that, for some fixed arbitrarily small $\tau>0$,
$F_{\ell}(a)\geq\ell(1-\tau)$ for $N_{0}\leq\ell\leq n$. We shall not need
this here.

$\displaystyle
q_{\ell}(a)=\nu_{\ell+1}(a)-(\nu_{\ell}(a)+p_{\ell}(a)+1)\,,\qquad\ell=0,\ldots,s_{j}(a)-1\,,$
$\displaystyle
q_{s_{j}(a)}(a)=\operatorname{max}\\{0,j-(\nu_{s_{j}(a)}(a)+p_{s_{j}(a)}(a)+1)\\}\,,\quad
F_{j}(a)=\sum_{\ell=0}^{s_{j}(a)}q_{\ell}(a).$

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)])

$\displaystyle|(T^{j-i}_{a})^{\prime}(T^{i+1}_{a}(c))|$
$\displaystyle=|(T^{j}_{a})^{\prime}(T_{a}(c))|$
$\displaystyle=|(T^{\nu_{1}-1}_{a})^{\prime}(T_{a}(c))|\cdot|(T^{j+1-\nu_{s_{j}}}_{a})^{\prime}(T^{\nu_{s}}_{a}(c))|$
$\displaystyle\qquad\cdot\biggl{(}\prod_{\ell=1}^{s_{j}-1}|(T^{p_{\ell}+1}_{a})^{\prime}(T^{\nu_{\ell}}_{a}(c))||(T^{q_{\ell}}_{a})^{\prime}(T^{\nu_{\ell}+p_{\ell}+1}_{a}(c))|\biggr{)}\,.$

Since $a$ satisfies $(BA)_{m}$ and $(FA)_{m}$ for all $m\leq n$, the bounds
(2.15) give $\lambda_{CE}$, $\gamma_{0}>0$, and $C_{0}>0$ such that

$\prod_{\ell=1}^{s_{j}-1}|(T^{p_{\ell}+1}_{a})^{\prime}(T^{\nu_{\ell}}_{a}(c))||(T^{q_{\ell}}_{a})^{\prime}(T^{\nu_{\ell}+p_{\ell}+1}_{a}(c))|\geq\prod_{\ell=1}^{s_{j}-1}C_{0}e^{\gamma_{0}q_{\ell}}\lambda_{CE}^{p_{\ell}/4}\,.$

Similarly,
$|(T^{\nu_{1}-1}_{a})^{\prime}(T_{a}(c))|>C_{0}e^{\gamma_{0}\nu_{1}}$. Next,
if $j\leq\nu_{s_{j}}+p_{s_{j}}+1$, we have

$|(T^{j+1-\nu_{s_{j}}}_{a})^{\prime}(T^{\nu_{s_{j}}}_{a}(c))|\geq
C_{0}^{2}\lambda_{BC}^{j-\nu_{s_{j}}}j^{-\kappa_{0}}e^{-r_{0}}\,,$

where we used (2.8) and [DMS, Lemma V.6.1.b, Prop. V.6.1]. If
$j>\nu_{s_{j}}+p_{s_{j}}+1$, we have, using [DMS, Lemma V.6.1.c, Prop. V.6.1]

$|(T^{j+1-\nu_{s_{j}}}_{a})^{\prime}(T^{\nu_{s}}_{a}(c))|\geq
C_{0}^{2}\lambda_{BC}^{p_{s_{j}}/4}e^{\gamma_{0}(j-(\nu_{s_{j}}-p_{s_{j}}-1))}e^{-r_{0}}\,,$

Summarising,

$|(T^{j}_{a})^{\prime}(T_{a}(c))|\geq\frac{C_{0}^{s_{j}+4}}{C}\lambda_{CE}^{(j-F_{j}(a))/4}e^{\gamma_{0}F_{j}(a)}j^{-\kappa_{0}}\,.$

Since $p_{\ell}\geq C_{0}r_{0}$ (see after (2.22)), we have $j-F_{j}\geq
jC_{0}r_{0}$ while $s_{j}\leq j/(C_{0}r_{0})$. We took $r_{0}$ large enough
(see after (2.15)) such that

(2.34) $C_{0}^{s_{j}+4}\lambda_{CE}^{(j-F_{j}(a))/4}\geq 1\,.$

Finally, using the trivial bound $e^{\gamma_{0}F_{j}(a)}\geq 1$, we find

$|(T^{j}_{a})^{\prime}(T_{a}(c))|\geq\frac{j^{-\kappa_{0}}}{C}\,.$

If $i\geq 1$ and $\nu_{\ell_{i}}(a)+p_{\ell_{i}}(a)<i<\nu_{\ell_{i}+1}(a)$ for
some $\ell_{i}\geq 1$, 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
$F_{i,j}(a)=\nu_{\ell_{i}+1}(a)-i+\sum_{\ell\geq\ell_{i}+1}^{s_{j}(a)}q_{\ell}(a)$.
Then

$|(T^{j-i}_{a})^{\prime}(T^{i+1}_{a}(c))|\geq\frac{C_{0}^{s_{j}-s_{i}+4}}{C}\lambda_{CE}^{((j-i)-F_{i,j}(a))/4}e^{\gamma_{0}F_{i,j}(a)}j^{-\kappa_{0}}\,.$

We have $j-i-F_{i,j}\geq(j-i)C_{0}r_{0}$ while
$s_{j}-s_{i}\leq(j-i)/(C_{0}r_{0})$, and we find, using
$e^{\gamma_{0}F_{i,j}(a)}\geq 1$ (we do not know or need
$F_{i,j}(a)\geq(1-\tau)(j-i)$),

$|(T^{j-i}_{a})^{\prime}(T^{i+1}_{a}(c))|\geq\frac{j^{-\kappa_{0}}}{C}\,.$

Otherwise, $\nu_{\ell_{i}}(a)\leq i-1\leq\nu_{\ell_{i}}(a)+p_{\ell_{i}}(a)$
for some $\ell_{i}\geq 1$. There may be (nonfree) 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 (2.11), we have
$r_{\ell_{i}}=O(\log(\nu_{\ell_{i}}))\leq C\kappa_{0}\log i$. Thus, the
missing factor in the $\ell$th bound period is $\leq\Lambda^{C\kappa_{0}\log
i}\leq i^{\kappa_{2}}$, and

$|(T^{j-i}_{a})^{\prime}(T^{i+1}_{a}(c))|\geq\frac{j^{-\kappa_{0}}}{Ci^{\kappa_{2}}}\,.$

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

Next, taking $a_{1},a_{2}\in\omega$, note that (2.7) (using the first bound of
(2.4) if $i<N_{0}$) implies that for all $N_{0}\leq i\leq n$, recalling
$T^{\prime}_{a}(x)=a(1-2x)$,

$\displaystyle|T_{a_{1}}^{\prime}(x_{i}(a_{1}))$ $\displaystyle-
T_{a_{2}}^{\prime}(x_{i}(a_{2}))|$
$\displaystyle\leq|T_{a_{1}}^{\prime}(x_{i}(a_{1}))-T_{a_{2}}^{\prime}(x_{i}(a_{1}))|+|T_{a_{2}}^{\prime}(x_{i}(a_{1}))-T_{a_{2}}^{\prime}(x_{i}(a_{2}))|$
(2.35)
$\displaystyle\leq|a_{1}-a_{2}|+2a_{2}|x_{i}(a_{1})-x_{i}(a_{2})|\leq(C+2a_{2})|x_{i}(\omega)|\,.$

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

(2.36)
$\biggl{|}\frac{(T_{a_{1}}^{j})^{\prime}(x_{0}(a_{1}))}{(T_{a_{2}}^{j})^{\prime}(x_{0}(a_{2}))}\biggr{|}\leq
1+C^{\prime\prime}e^{C^{\prime\prime}}j^{2\kappa_{0}+1+\kappa_{2}}|x_{j}(\omega)|\,,\,\forall
1\leq j\leq n\,.$

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

$\prod_{i=0}^{j-1}(1+\upsilon_{i})\leq\exp\biggl{(}\sum_{i=0}^{j-1}\upsilon_{i}\biggr{)}\leq
1+e^{\sum_{i=0}^{j-1}\upsilon_{i}}\sum_{i=0}^{j-1}\upsilon_{i}\,,\,\,\mbox{ if
all }\upsilon_{i}\geq 0\,,$

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

$\displaystyle\biggl{|}\frac{(T_{a_{1}}^{j})^{\prime}(x_{0}(a_{1}))}{(T_{a_{2}}^{j})^{\prime}(x_{0}(a_{2}))}$
$\displaystyle\biggr{|}=\biggl{|}\prod_{i=0}^{j-1}\frac{T_{a_{1}}^{\prime}(x_{i}(a_{1}))}{T_{a_{2}}^{\prime}(x_{i}(a_{2}))}\biggr{|}$
$\displaystyle\leq
1+e^{\sum_{i}\Bigl{|}-1+\frac{T_{a_{1}}^{\prime}(x_{i}(a_{1}))}{T_{a_{2}}^{\prime}(x_{i}(a_{2}))}\Bigr{|}}\cdot\sum_{i=0}^{j-1}\biggl{|}\frac{T_{a_{1}}^{\prime}(x_{i}(a_{1}))}{T_{a_{2}}^{\prime}(x_{i}(a_{2}))}-1\biggr{|}$
$\displaystyle\leq
1+e^{(C+2a_{2})\sum_{i=0}^{j-1}C|x_{i}(\omega)|i^{\kappa_{0}}}\cdot(C+2a_{2})\sum_{i=0}^{j-1}C|x_{i}(\omega)|i^{\kappa_{0}}$
(2.37) $\displaystyle\leq
1+e^{C^{\prime\prime}j^{2\kappa_{0}+1+\kappa_{2}}|x_{j}(\omega)|}\cdot
C^{\prime\prime}j^{2\kappa_{0}+1+\kappa_{2}}|x_{j}(\omega)|\,,\,\,\forall
j\leq n\,,$

where we used (2.35) and (2.8) (the first bound of (2.4) if $i<N_{0}$) in the
second inequality, and (2.32) in the last inequality. Setting
$M_{0}:=4\kappa_{0}+3+2\kappa_{2}$, if (2.30) holds for $\omega$, then (2.32)
gives for all $N_{0}\leq j\leq n$

$\displaystyle C^{\prime\prime}j^{2\kappa_{0}+1+\kappa_{2}}|x_{j}(\omega)|$
$\displaystyle\leq
C^{\prime\prime}j^{2\kappa_{0}+1+\kappa_{2}}C^{\prime}j^{\kappa_{0}+1+\kappa_{2}}|x_{n}(\omega)|$
$\displaystyle\leq
C^{\prime}\frac{j^{3\kappa_{0}+3+2\kappa_{2}}}{n^{M_{0}/(1-\alpha)}}\leq
C^{\prime\prime\prime}\,.$

This proves (2.36). Similarly,
$\Bigl{|}\frac{(T_{a_{2}}^{j})^{\prime}(x_{0}(a_{2}))}{(T_{a_{1}}^{j})^{\prime}(x_{0}(a_{1}))}\Bigr{|}\leq
1+C^{\prime\prime}e^{C^{\prime\prime}}j^{2\kappa_{0}+1+\kappa_{2}}|x_{j}(\omega)|$.
Therefore,

$\biggl{|}\frac{1}{(T_{a_{1}}^{j})^{\prime}(x_{0}(a_{1}))}-\frac{1}{(T_{a_{2}}^{j})^{\prime}(x_{0}(a_{2}))}\biggr{|}\leq
C^{\prime\prime\prime}\frac{j^{2\kappa_{0}+1+\kappa_{2}}|x_{j}(\omega)|}{|(T_{a_{1}}^{j})^{\prime}(x_{0}(a_{1}))|}\,.$

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

$\frac{x_{n}^{\prime}(a)}{(T_{a}^{n})^{\prime}(x_{0}(a))}=x_{0}^{\prime}(a)+\sum_{j=1}^{n}\frac{(\partial_{a}T_{a})(x_{j-1}(a))}{(T_{a}^{j})^{\prime}(x_{0}(a))}\,,\,\forall
a\in\tilde{\omega}\in\mathcal{P}_{n}\,,$

for $a=a_{1}$ and $a=a_{2}$, we find, since
$x_{0}^{\prime}(a)=\partial_{a}c_{1}(a)=1/4$, and

$|\partial_{a}T_{a}|_{a_{1}}(x_{j-1}(a_{1}))-\partial_{a}T_{a}|_{a_{2}}(x_{j-1}(a_{2}))|\leq|x_{j-1}(a_{1})-x_{j-1}(a_{2})|\leq|x_{j-1}(\omega)|\,,$

recalling (2.5), and applying (2.32) and then (2.30) for
$M_{0}=4\kappa_{0}+3+2\kappa_{2}$,

$\displaystyle\biggl{|}\frac{x_{n}^{\prime}(a_{1})}{(T_{a_{1}}^{n})^{\prime}(x_{0}(a_{1}))}\biggr{|}$
$\displaystyle\leq\biggl{|}\frac{x_{n}^{\prime}(a_{2})}{(T_{a_{2}}^{n})^{\prime}(x_{0}(a_{2}))}\biggr{|}+\hat{C}|a_{1}-a_{2}|+\hat{C}\sum_{j=1}^{n}\frac{j^{2\kappa_{0}+1+\kappa_{2}}}{\lambda_{CE}^{j}}|x_{j}(\omega)|$
$\displaystyle\leq\biggl{|}\frac{x_{n}^{\prime}(a_{2})}{(T_{a_{2}}^{n})^{\prime}(x_{0}(a_{2}))}\biggr{|}+\bar{C}C^{\prime}n^{4\kappa_{0}+3+2\kappa_{2}}|x_{n}(\omega)|$
$\displaystyle\leq\biggl{|}\frac{x_{n}^{\prime}(a_{2})}{(T_{a_{2}}^{n})^{\prime}(x_{0}(a_{2}))}\biggr{|}+\tilde{C}|x_{n}(\omega)|^{\alpha}\,.$

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

$\displaystyle\biggl{|}\frac{x_{n}^{\prime}(a_{1})}{x_{n}^{\prime}(a_{2})}\biggr{|}$
$\displaystyle\leq\biggl{|}\frac{(T_{a_{1}}^{n})^{\prime}(x_{0}(a_{1}))}{(T_{a_{2}}^{n})^{\prime}(x_{0}(a_{2}))}\biggr{|}\biggl{(}1+\tilde{C}|x_{n}(\omega)|^{\alpha}\frac{|(T_{a_{2}}^{n})^{\prime}(x_{0}(a_{2}))|}{|x_{n}^{\prime}(a_{2})|}\biggr{)}$
$\displaystyle\leq 1+C\tilde{C}|x_{n}(\omega)|^{\alpha}\,.\qed$

###  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 $a\in\Omega_{*}$
satisfies242424The factor $\|\varphi\|_{L_{1}(dm)}$ in the right-hand side of
[Sch, Prop. 4.3] is replaced in Proposition 2.5 by
$\|\varphi\|_{L^{1}(d\mu_{a})}\leq\|\varphi\|_{L^{\infty}(dm)}$. This does not
impact [Sch, p. 36, use of Prop. 4.3]. the following uniform decorrelation
result for Hölder continuous observables. 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 $\varphi\in H^{s}_{q}(I)$, $\psi\in L^{\infty}(dm)$,
$a\in\Omega_{*}(a_{*},\kappa_{0})$

$\biggl{|}\int_{0}^{1}\varphi(\psi\circ
T_{a}^{n})\,dm-\int_{0}^{1}\varphi\,dm\int_{0}^{1}\psi\,d\mu_{a}\biggr{|}\leq
C\|\varphi\|_{H^{s}_{q}}\|\psi\|_{L^{1}(d\mu_{a})}(\rho^{s}_{q})^{n}\,,\forall
n\geq 1\,.$

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

$\biggl{|}\int_{0}^{1}\varphi(\psi\circ
T_{a}^{n})\,d\mu_{a}-\int_{0}^{1}\varphi\,d\mu_{a}\,\int_{0}^{1}\psi\,d\mu_{a}\biggr{|}\leq
C\|\varphi\|_{\varpi}\|\psi\|_{L^{1}(d\mu_{a})}(\rho_{\varpi})^{n}\,,\,\forall
n\geq 1\,.$

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)
$\biggl{|}\int\varphi\,d\mu_{a}-\int\varphi\,d\mu_{a^{\prime}}\biggr{|}\leq
C|a-a^{\prime}|^{\Theta}\|\varphi\|_{1/2}\,,\quad\forall
a,a^{\prime}\in\Omega_{*}(a_{*},\kappa_{0})\,.$

Our proof of Proposition 2.5 uses the following facts.

######  Sublemma 2.7.

For any $a\in\Omega_{BC}$, 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 $\tilde{a}$, there exists
$C_{s,q,\tilde{a}}<\infty$ such that

$\sup_{a\in\Omega_{*}{a_{*},\kappa_{0}}}\|h_{a}\|_{H^{s}_{q}(I)}\leq
C_{s,q,\tilde{a}}\,.$

###### 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 $a\in\Omega_{BC}$, set
$T^{-k}_{a,\varsigma}:=(T_{a}^{k}|_{U_{k,a,\varsigma}})^{-1}$, for $k\geq 1$
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
$\lambda_{CE}>e^{4\alpha_{BC}}$ in the proof of Proposition 2.2, use [BS1,
Prop 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)
$h_{a}(x)=\psi_{a}(x)+\sum_{k=1}^{\infty}\sum_{\varsigma\in\\{+,-\\}}\chi_{k,a}(x)\frac{\Xi^{k}_{a,\varsigma}(T^{-k}_{a,\varsigma}(x))\psi_{a}(T^{-k}_{a,\varsigma}(x))}{|(T_{a}^{k})^{\prime}(T^{-k}_{a,\varsigma}(x))|}\,,$

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)
$\frac{\Psi(T^{-k}_{a,\varsigma}(x))}{|(T_{a}^{k})^{\prime}(T^{-k}_{a,\varsigma}(x))|}=\frac{\Psi_{k,1}(x)}{|x-c_{k}(a)|^{1/2}}+\Psi_{k,2}(x)|x-c_{k}(a)|^{1/2}+\Psi_{k,3}(x)\,,$

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

For the second claim, it is convenient to use an alternative decomposition of
$h_{a}$. First recall that [BBS, Cor 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
$H_{0}\geq 1$ and a set
$\Delta_{0}(\tilde{a},\kappa_{0})\subset\Omega_{T}\cap\Omega_{\mathrm{slow}}$
of $(H_{0},\kappa_{0})$-polynomially recurrent 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,
Prop. 2.1], and it suffices to observe that the required uniformity in
constants is satisfied by (2.5) and (2.8), while [BBS, (8) and (7)] are
exactly [DMS, V.(6.1), V.(6.2) in Prop. V.6.1].

Let then

$\Pi_{a}(\hat{\psi})(x)=\sum_{j\geq
0,\varsigma\in\pm}\frac{\lambda^{j}}{|(T_{a}^{j})^{\prime}(T^{-j}_{a,\varsigma}(x))|}\psi_{j}(T^{-j}_{a,\varsigma}(x))\,,$

(for a suitable $\lambda>1$) 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
$\mathcal{L}_{a}\varphi(x)=\sum_{T_{a}(y)=x}\varphi(y)/|T^{\prime}_{a}(y)|$.
Then252525[BBS, (66)] gives uniform Lasota–Yorke estimates. [BBS, Lemma 3.8,
Lemma 4.5, Lemma 4.6, Prop. 4.1] give the weak norm bounds needed for
Keller–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)
$\|\hat{\mathcal{L}}_{a}^{n}(\hat{\varphi})-\hat{h}_{a}\hat{\nu}(\hat{\varphi})\|^{\prime}_{a}\leq
C\|\hat{\varphi}\|^{\prime}_{a}\theta^{n}\,,\qquad\forall\hat{\varphi}\in\mathcal{B}_{a}^{W^{1}_{1}}\,,\quad\forall
a\in\Omega_{\tilde{a}}\,,$

where $\hat{h}_{a}$ is the fixed point262626The fixed point property
determines $\hat{h}_{a}$ by its value on the level zero of the tower. of
$\hat{\mathcal{L}}_{a}$ on $\mathcal{B}_{a}^{W^{1}_{1}}$ normalised by
$\int\Pi_{a}\hat{h}_{a}\,dm=1$, while $\hat{\nu}$, the nonnegative measure
whose density with respect to Lebesgue in the level $j$ of the tower is
$\lambda^{j}$, is the fixed point of the dual of $\hat{\mathcal{L}}_{a}$ (see
[BS1, (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, Chapter 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 (2.40) combined with the uniform bound (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 11–12]. ∎

###### Proof of Proposition 2.5.

Recall from the proof of Proposition 2.2 that we have
$\lambda_{CE}>e^{4\alpha_{BC}}$. 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 $\varphi\in C^{1}$: 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, Cor. 1.6] by $\Omega_{*}(a_{*},\kappa_{0})$. The
first bound for Lipschitz continuous $\varphi$ thus follows from the second
assertion of [BBS, Cor. 1.6], since $\Omega_{*}\subset[a_{\mathrm{mix}},4)$.
Indeed, note first that $a$ is topologically mixing if and only if its
renormalisation period $P_{a}$ is equal to one. Second, observe that the
constant $C_{\varphi,\psi}$ in the second claim of [BBS, Cor. 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, Lemma 4.5, Lemma 4.6] and the principle of
uniform boundedness. More precisely, using the notation from the proof of
Sublemma 2.7, we have

$\displaystyle\int(\psi\circ
T_{a}^{n})\varphi\,dm=\int\psi\Pi_{a}(\hat{\mathcal{L}}_{a}^{n}(\hat{\varphi}))\,dm\qquad\qquad\mbox{
if }\Pi_{a}(\hat{\varphi})=\varphi\,,$ $\displaystyle\int(\psi\circ
T_{a}^{n})\varphi
h_{a}\,dm=\int\psi\Pi_{a}(\hat{\mathcal{L}}_{a}^{n}(\hat{\varphi}_{a}))\,dm\qquad\,\,\mbox{
if }\Pi_{a}(\hat{\varphi}_{a})=\varphi h_{a}\,.$

Since272727The 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
$a=4$.  $\Pi_{a}\hat{h}_{a}=h_{a}$, 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 $j\geq 1$) such that, on
the one hand, $\|\hat{\varphi}\|_{a}^{\prime}\leq C\|\varphi\|_{1}$ uniformly
in $a$, and, on the other hand, $\hat{\nu}(\hat{\varphi})=\int\varphi\,dm$, we
conclude by applying (2.41) from the proof of Sublemma 2.7. ∎

###### Proof of Proposition 2.6.

If $a=a_{*}$, the bound is an immediate consequence of the first claim of
[BBS, Cor. 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 $a\neq a_{*}$, 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

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

and such that for all
$a,a^{\prime}\in\Omega_{*}(a_{*},\kappa_{0})\cap[a_{*}-\epsilon_{\varphi},a_{*}+\epsilon_{\varphi}]$
we have

(2.42)
$|\sigma_{a}(\varphi)-\sigma_{a^{\prime}}(\varphi)|\leq\frac{C}{2C_{\epsilon}(\varphi)}\|\varphi\|_{\varpi}|a-a^{\prime}|^{\theta}\,.$

###### 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

$\sum_{k>k_{0}}\biggl{|}\int\biggl{(}\varphi-\int\varphi\,d\mu_{a}\biggr{)}\cdot\biggl{(}\biggl{(}\varphi-\int\varphi\,d\mu_{a}\biggr{)}\circ
T_{a}^{k}\biggr{)}\,d\mu_{a}\biggr{|}\\\ \leq
C_{0}\|\varphi\|_{\varpi}^{2}\cdot\frac{\rho^{k_{0}}}{1-\rho}\,,\qquad\forall
k_{0}\geq 1\,,\quad\forall a\in\Omega_{a_{*}}\,,\quad\forall\varphi\in
C^{\varpi}\,.$

Since $\int\bigl{(}(\varphi-A_{a})\circ T_{a}^{k}\bigr{)}(\varphi-
A_{a})\,d\mu_{a}=\int(\varphi\circ T_{a}^{k})\varphi\,d\mu_{a}-A_{a}^{2}$, we
have

$\displaystyle|\sigma_{a}(\varphi)^{2}-\sigma_{a^{\prime}}(\varphi)^{2}|\leq
2$ $\displaystyle\sum_{k=0}^{k_{0}-1}\biggl{|}\int\varphi(\varphi\circ
T_{a}^{k})\,d\mu_{a}-\int\varphi(\varphi\circ
T_{a^{\prime}}^{k})\,d\mu_{a}\biggr{|}$
$\displaystyle+2\sum_{k=0}^{k_{0}-1}\biggl{|}\int\varphi(\varphi\circ
T_{a^{\prime}}^{k})\,d\mu_{a}-\int\varphi(\varphi\circ
T_{a^{\prime}}^{k})\,d\mu_{a^{\prime}}\biggr{|}$
$\displaystyle+2\sum_{k=0}^{k_{0}-1}\biggl{|}\biggl{(}\int\varphi\,d\mu_{a}\biggr{)}^{2}-\biggl{(}\int\varphi\,d\mu_{a^{\prime}}\biggr{)}^{2}\biggr{|}$
$\displaystyle+4C_{0}\|\varphi\|^{2}_{\varpi}\frac{\rho^{k_{0}}}{1-\rho}\,,\qquad\forall
k_{0}\geq 1\,.$

Assume for a moment that $\varpi\geq 1/2$. 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 $\varphi\in C^{\varpi}$

$|\sigma_{a}(\varphi)^{2}-\sigma_{a^{\prime}}(\varphi)^{2}|\leq
k_{0}C_{1}\|\varphi\|^{2}_{\varpi}\Lambda^{k_{0}}|a-a^{\prime}|^{\Theta}+C_{0}\|\varphi\|^{2}_{\varpi}\frac{\rho^{k_{0}}}{1-\rho}\\\
+2\sum_{k=0}^{k_{0}-1}\biggl{|}\int\varphi(\varphi\circ
T_{a}^{k})\,d\mu_{a}-\int\varphi(\varphi\circ
T_{a^{\prime}}^{k})\,d\mu_{a}\biggr{|}\,,\qquad\forall k_{0}\geq 1.$

Next, (2.2) gives that

$\displaystyle\int$ $\displaystyle|\varphi\circ T_{a}^{k}-\varphi\circ
T_{a^{\prime}}^{k}|\,d\mu_{a}\leq\|\varphi\|_{\varpi}(C\Lambda^{k}|a-a^{\prime}|)^{\varpi}\,.$

Therefore, we find

(2.43)
$\bigl{|}\sigma_{a}(\varphi)^{2}-\sigma_{a^{\prime}}(\varphi)^{2}\bigr{|}\leq
k_{0}C_{1}\|\varphi\|^{2}_{\varpi}\Lambda^{k_{0}}|a-a^{\prime}|^{\Theta}+4C_{0}\|\varphi\|^{2}_{\varpi}\frac{\rho^{k_{0}}}{1-\rho}\\\
+k_{0}\|\varphi\|_{\varpi}(C\Lambda^{k_{0}}|a-a^{\prime}|)^{\varpi}\,.$

We conclude the proof for $\varpi\geq 1/2$ by dividing (2.43) by
$|a-a^{\prime}|^{\theta}$, for small enough $\theta>0$ and optimising 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

$\|\varphi_{\upsilon}\|_{1/2}\leq
C_{4}\upsilon^{\varpi-1/2}\|\varphi\|_{\varpi}\,,\,\,\sup|(\varphi\circ
T_{\bar{a}}^{k})\varphi-(\varphi_{\upsilon}\circ
T_{\bar{a}}^{k})\varphi_{\upsilon}|\leq
C_{4}\upsilon^{\varpi}\Lambda^{k}\|\varphi\|_{\varpi}\,,$

for all small $\upsilon>0$, all $0\leq k\leq k_{0}$, and all
$\bar{a}\in\Omega_{a_{*}}$. To conclude, optimise 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 $a_{*}$, $\kappa_{0}\geq 3/d_{1}(a_{*})$,
$\mathcal{P}_{j}(a_{*},\kappa_{0})$, and
$\Omega_{*}=\Omega_{*}(a_{*},\kappa_{0})$ be as in Proposition 2.2 and fix
$\varpi\in(0,1)$. This section is devoted to Proposition 3.2, the main
estimate (analogous to [Sch, Prop. 5.1]) towards a law of large numbers for
the squares of the blocks which will be defined in Section 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 (1.6), (1.3), and assume
$\sigma_{a_{*}}(\varphi)>0$. Lemma 2.8 gives $\epsilon_{\varphi}>0$ such that

(3.1) $\sigma_{a}(\varphi)>0\,,\quad\forall
a\in\Omega_{*}^{\varphi}:=\Omega_{*}\cap[a_{*}-\epsilon_{\varphi},a_{*}+\epsilon_{\varphi}]\,.$

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 $a\mapsto\sigma_{a}(\varphi)\geq 0$ is $\theta$-Hölder continuous
on $\Omega_{*}$ for some $\theta<\operatorname{min}\\{1/2,\varpi\\}$, and
uniformly bounded away from zero on $\Omega_{*}^{\varphi}$. Taking
$\Theta\geq\theta$, 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)
$\|\varphi_{a}\|_{\infty}\leq\|\varphi_{a}\|_{\varpi}\leq\widetilde{C}\|\varphi\|_{\varpi}\,,\qquad\forall
a\in[a_{*}-\epsilon_{\varphi},a_{*}+\epsilon_{\varphi}]\,.$

Then, using (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)
$\xi_{n}(a)-\xi_{n}(a^{\prime})=\varphi_{a}(x_{n}(a))-\varphi_{a^{\prime}}(x_{n}(a))+\varphi_{a^{\prime}}(T_{a}^{n+1}(c))-\varphi_{a^{\prime}}(T_{a^{\prime}}^{n+1}(c))\,.$

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

(3.4) $\frac{M_{0}}{1-\alpha}\leq\frac{3}{\alpha}\,.$

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

(3.5) $\lambda_{0}=\operatorname{min}(\lambda_{CE}^{\theta},\rho^{-1/2})>1\,,$

where $\lambda_{CE}>1$ is given by (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 (2.1), let
$\eta\in(0,1/2)$ be so small that

(3.6)
$\Bigl{(}\frac{2\Lambda}{\lambda_{CE}}\Bigr{)}^{\eta}\leq\lambda_{0}<\frac{\lambda_{CE}^{\varpi}}{\Lambda^{\eta\varpi}}\,.$

Define the expectation $E(\psi)$ of $\psi\in L_{\infty}(I)$ by282828We
restrict to the Cantor set $\Omega_{*}^{\varphi}$ here and thus in (3.8). The
bound (2.9) is used in the proof of (3.8) (but not for (3.9), Lemma 3.3, or
Corollary 3.4).

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

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)
$\Biggl{|}E\Biggl{(}\sum_{j=k}^{k+n-1}\xi_{j}\Biggr{)}^{2}-n\Biggr{|}\leq
C_{\varphi}\,,\quad\forall k\geq\operatorname{min}\\{2K,[2/\eta]\\}\,,\
\forall 1\leq n\leq\eta k/2\,,$

and, setting292929The 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 nontrivial
interval $\omega\subset\tilde{\omega}\in\mathcal{P}_{v(k)}$ with
$\omega\cap\Omega_{*}^{\varphi}\neq\emptyset$ and
$\lambda_{0}^{-k^{1/4}}\leq|x_{v(k)}(\omega)|\leq n^{-3/\alpha}$, we have

(3.9)
$\biggl{|}\frac{1}{|\omega|}\int_{\omega}\biggl{(}\sum_{j=k}^{k+n-1}\xi_{j}\biggr{)}^{2}\,dm-n\biggr{|}\leq
C_{\varphi}\,,\quad\forall k\geq[2/\eta]\,,\ \forall 1\leq n\leq\eta k/2\,.$

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

$n\leq n_{i}\leq n+\eta n\,,\qquad 1\leq i\leq\ell_{0}\,,$

for every $\tilde{\omega}\in\mathcal{P}_{n}$ and each nontrivial interval
$\omega\subset\tilde{\omega}$ with
$\omega\cap\Omega_{*}^{\varphi}\neq\emptyset$,

(3.10)
$\displaystyle\int\limits_{x_{n}(\omega)}\biggl{|}\prod_{\ell=1}^{\ell_{0}}\xi_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y))$
$\displaystyle-\prod_{\ell=1}^{\ell_{0}}\varphi_{a_{0}}(T_{a_{0}}^{n_{\ell}-n}(y))\biggr{|}\,dy$
$\displaystyle\leq C\lambda_{0}^{-n}|x_{n}(\omega)|\,,\qquad\forall
a_{0}\in\omega\cap\Omega_{*}^{\varphi}\,.$

######  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,
$|x_{n}(\omega)|\leq n^{-3/\alpha}$, then for any
$a_{0}\in\omega\cap\Omega_{*}^{\varphi}$,

$\biggl{|}\frac{1}{|\omega|}\int_{\omega}\prod_{\ell=1}^{\ell_{0}}\xi_{n_{\ell}}(a)\,da-\frac{1}{|x_{n}(\omega)|}\int_{x_{n}(\omega)}\prod_{\ell=1}^{\ell_{0}}\varphi_{a_{0}}(T_{a_{0}}^{n_{\ell}-n}(y))\,dy\biggr{|}\\\
\leq C_{3}(|x_{n}(\omega)|^{\alpha}+\lambda_{0}^{-n})\,.$

###### Proof.

Since (3.4) implies (2.30) for $\omega$, the change of variables $y=x_{n}(a)$
on $\omega$, combined with the distortion estimate (2.31), gives

$\displaystyle\biggl{|}\frac{1}{|\omega|}$
$\displaystyle\int_{\omega}\prod_{\ell=1}^{\ell_{0}}\xi_{n_{\ell}}(a)\,da-\frac{1}{|x_{n}(\omega)|}\int_{x_{n}(\omega)}\prod_{\ell=1}^{\ell_{0}}\xi_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y))\,dy\biggr{|}$
(3.11)
$\displaystyle=\frac{1}{|x_{n}(\omega)|}\biggl{|}\int_{x_{n}(\omega)}\prod_{\ell=1}^{\ell_{0}}\xi_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y))\biggl{(}\frac{1}{|x_{n}^{\prime}(x_{n}|_{\omega}^{-1}y)|}\frac{|x_{n}(\omega)|}{|\omega|}-1\biggr{)}dy\biggr{|}$
$\displaystyle\leq
C\frac{|x_{n}(\omega)|^{\alpha}}{|x_{n}(\omega)|}\int_{x_{n}(\omega)}\prod_{\ell=1}^{\ell_{0}}|\xi_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y))|\,dy\,.$

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

$\tilde{\varphi}_{\ell}(y)=\tilde{\varphi}_{\ell,a_{0}}(y)=\varphi_{a_{0}}(T_{a_{0}}^{n_{\ell}-n}(y))\,,\qquad\tilde{\xi}_{\ell}(y)=\tilde{\xi}_{\ell,\omega}(y)=\xi_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y))$

with

$\xi_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y))=\varphi_{x_{n}|_{\omega}^{-1}(y)}(x_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y))=\varphi_{x_{n}|_{\omega}^{-1}(y)}(T^{n_{\ell}+1}_{x_{n}|_{\omega}^{-1}(y)}(c))$

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

$\displaystyle|\tilde{\xi}_{1}\tilde{\xi}_{2}\tilde{\xi}_{3}\tilde{\xi}_{4}-\tilde{\varphi}_{1}\tilde{\varphi}_{2}\tilde{\varphi}_{3}\tilde{\varphi}_{4}|\leq|(\tilde{\xi}_{1}$
$\displaystyle-\tilde{\varphi}_{1})\tilde{\xi}_{2}\tilde{\xi}_{3}\tilde{\xi}_{4}|+|\tilde{\varphi}_{1}(\tilde{\xi}_{2}-\tilde{\varphi}_{2})\tilde{\xi}_{3}\tilde{\xi}_{4}|$
(3.12)
$\displaystyle+|\tilde{\varphi}_{1}\tilde{\varphi}_{2}(\tilde{\xi}_{3}-\tilde{\varphi}_{3})\tilde{\xi}_{4}|+|\tilde{\varphi}_{1}\tilde{\varphi}_{2}\tilde{\varphi}_{3}(\tilde{\xi}_{4}-\tilde{\varphi}_{4})|\,,$

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

$\frac{1}{|x_{n}(\omega)|}\int_{x_{n}(\omega)}|\tilde{\xi}_{\ell,\omega}-\tilde{\varphi}_{\ell,a_{0}}|\,dy\leq\bar{C}\lambda_{0}^{-n}\,,\qquad\forall
a_{0}\in\omega\cap\Omega_{*}^{\varphi}\,,\,1\leq\ell\leq\ell_{0}\,.$

We will do so by showing the pointwise estimate

$|\tilde{\xi}_{\ell,\omega}(y)-\tilde{\varphi}_{\ell,a_{0}}(y)|\leq\bar{C}\lambda_{0}^{-n}\,,\qquad\forall
y\in x_{n}(\omega)\,,\,\forall
a_{0}\in\omega\cap\Omega_{*}^{\varphi}\,,\,1\leq\ell\leq\ell_{0}\,.$

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

$\displaystyle\tilde{\xi}_{\ell,\omega}(y)$
$\displaystyle-\tilde{\varphi}_{\ell,a_{0}}(y)=\xi_{n_{\ell}}(a)-\varphi_{a_{0}}(T_{a_{0}}^{n_{\ell}-n}(x_{n}(a)))$
$\displaystyle=\varphi_{a}(x_{n_{\ell}}(a))-\varphi_{a_{0}}(T_{a_{0}}^{n_{\ell}-n}(x_{n}(a)))$
(3.13)
$\displaystyle=\varphi_{a}(x_{n_{\ell}}(a))-\varphi_{a_{0}}(x_{n_{\ell}}(a))+\varphi_{a_{0}}(x_{n_{\ell}}(a))-\varphi_{a_{0}}(T_{a_{0}}^{n_{\ell}-n}(x_{n}(a)))\,.$

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

(3.14) $|\varphi_{a}(x_{n_{\ell}}(a))-\varphi_{a_{0}}(x_{n_{\ell}}(a))|\leq
C|\omega|^{\theta}\,,\qquad\forall\\{a,a_{0}\\}\subset\omega\,.$

Hence, using our choice (3.5) of $\lambda_{0}$, and since $|\omega|\leq
C\lambda_{CE}^{-n}$ by (2.7), we get

(3.15) $|\varphi_{a}(x_{n_{\ell}}(a))-\varphi_{a_{0}}(x_{n_{\ell}}(a))|\leq
C|\omega|^{\theta}\leq C\lambda_{0}^{-n}.$

For the last two terms in the right-hand side of (3.13) 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 (2.2),

(3.16)
$\bigl{|}x_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y))-T_{a_{0}}^{n_{\ell}-n}(y)\bigr{|}=|T_{a}^{n_{\ell}-n}(y)-T_{a_{0}}^{n_{\ell}-n}(y)|\\\
\leq C\Lambda^{n_{\ell}-n}|a-a_{0}|\leq
C\Lambda^{n_{\ell}-n}|\omega|\,,\qquad\forall y\in x_{n}(\omega)\,.$

Then, since $n_{\ell}-n\leq\eta n$, our choice of $\lambda_{0}$, $\eta$, with
(3.2) at $a=a_{0}$ give303030We do not need the analogue of Sublemma 5.4 from
[Sch] here.

(3.17) $\displaystyle|\varphi_{a_{0}}(x_{n_{\ell}}(a))$
$\displaystyle-\varphi_{a_{0}}(T_{a_{0}}^{n_{\ell}-n}(x_{n}(a)))|$
$\displaystyle=\bigl{|}\varphi_{a_{0}}(x_{n_{\ell}}(x_{n}|_{\omega}^{-1}(y)))-\varphi_{a_{0}}(T_{a_{0}}^{n_{\ell}-n}(y))\bigr{|}$
$\displaystyle\leq C\tilde{C}\Lambda^{(n_{\ell}-n)\varpi}|\omega|^{\varpi}\leq
C\tilde{C}\Lambda^{\varpi\eta n}|\omega|^{\varpi}\leq
C\tilde{C}\lambda_{0}^{-n}\,,$

using again in the last inequality that $|\omega|\leq C\lambda_{CE}^{-n}$ from
(2.7). We conclude by combining (3.15) and (3.17) into (3.13). ∎

###  3.1. Proof of Proposition 3.2

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

$\int_{\omega}\biggl{(}\sum_{j=k}^{k+n-1}\xi_{j}\biggr{)}^{2}\,dm=\sum_{j=k}^{k+n-1}\biggl{(}\int_{\omega}\xi_{j}^{2}\,dm+2\sum_{\ell=j+1}^{k+n-1}\int_{\omega}\xi_{j}\xi_{\ell}\,dm\biggr{)}\,,$

it is sufficient to show that

(3.18)
$\sum_{j=k}^{k+n-1}\biggl{|}1-\frac{1}{|\omega|}\int_{\omega}\biggl{(}\xi_{j}^{2}+2\sum_{\ell=j+1}^{k+n-1}\xi_{j}\xi_{\ell}\biggr{)}\,dm\biggr{|}=O(1)\,.$

Fix $a_{0}\in\omega\cap\Omega_{*}^{\varphi}$. By Corollary 3.4 for
$\ell_{0}=2$, we have, for $k\leq j\leq k+n-1$,

$\frac{1}{|\omega|}\int_{\omega}\biggl{(}\xi_{j}^{2}+2\sum_{\ell=j+1}^{k+n-1}\xi_{j}\xi_{\ell}\biggr{)}\,dm\\\
=\frac{1}{|x_{v}(\omega)|}\int_{x_{v}(\omega)}\biggl{(}\varphi_{a_{0}}^{2}\circ
T_{a_{0}}^{j-v}+2\sum_{\ell=j+1}^{k+n-1}\varphi_{a_{0}}\circ
T_{a_{0}}^{j-v}\varphi_{a_{0}}\circ T_{a_{0}}^{\ell-v}\biggr{)}\,dm\\\
+O\bigl{(}(k+n-j)(\lambda_{0}^{-(k-k^{1/4})}+|x_{v}(\omega)|^{\alpha})\bigr{)}\,,$

(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

$\displaystyle\int_{x_{v}(\omega)}(\varphi_{a_{0}}\circ T_{a_{0}}^{j-v}$
$\displaystyle)(\varphi_{a_{0}}\circ T_{a_{0}}^{\ell-v})\,dm$
$\displaystyle=|x_{v}(\omega)|\int\varphi_{a_{0}}\cdot(\varphi_{a_{0}}\circ
T_{a_{0}}^{\ell-j})\,d\mu_{a_{0}}+O(\rho^{j-v})\,,\quad\forall\ell\geq j\,.$

Hence,

$\frac{1}{|\omega|}\int_{\omega}\biggl{(}\xi_{j}^{2}+2\sum_{\ell=j+1}^{k+n-1}\xi_{j}\xi_{\ell}\biggr{)}\,dm=\int\biggl{(}\varphi_{a_{0}}^{2}+2\sum_{\ell=k+1}^{k+n-1}\varphi_{a_{0}}\cdot(\varphi_{a_{0}}\circ
T_{a_{0}}^{\ell-j})\biggr{)}\,d\mu_{a_{0}}\\\
+O\bigl{(}(k+n-j)(\lambda_{0}^{-(k-k^{1/4})}+|x_{v}(\omega)|^{\alpha})\bigr{)}+O(\rho^{j-v}|x_{v}(\omega)|^{-1})\,.$

By (1.3) and (1.7), we have

$1=\int\varphi_{a_{0}}^{2}\,d\mu_{a_{0}}+2\sum_{i=1}^{\infty}\int\varphi_{a_{0}}\cdot\varphi_{a_{0}}\circ
T_{a_{0}}^{i}\,d\mu_{a_{0}}\,.$

Therefore, the second claim of Proposition 2.5 gives

$\int\biggl{(}\varphi_{a_{0}}^{2}+2\sum_{\ell=j+1}^{k+n-1}\varphi_{a_{0}}\cdot(\varphi_{a_{0}}\circ
T_{a_{0}}^{\ell-j})\biggr{)}\,d\mu_{a_{0}}=1+O(\rho^{k+n-j})\,.$

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

$\biggl{|}1-\frac{1}{|\omega|}\int_{\omega}\biggl{(}\xi_{j}^{2}+2\sum_{\ell=j+1}^{k+n-1}\xi_{j}\xi_{\ell}\biggr{)}\biggr{|}\\\
\leq
C(k+n-j)\bigl{(}\lambda_{0}^{-(k-k^{1/4})}+|x_{v}(\omega)|^{\alpha}+\rho^{j-v}|x_{v}(\omega)|^{-1}\bigr{)}+C\rho^{k+n-j}\,.$

To proceed we shall use several times that

$\sup_{n}\sup_{k}\sum_{j=k}^{k+n-1}\frac{1}{(k+n-j)^{2}}\leq\sup_{n}\sum_{\ell=1}^{n}\frac{1}{\ell^{2}}<\infty\,.$

Clearly, $\rho^{k+n-j}\leq\frac{C}{(k+n-j)^{2}}$. For the term
$(k+n-j)\rho^{j-v}|x_{v}(\omega)|^{-1}$, we use
$|x_{v}(\omega)|\geq\lambda_{0}^{-k^{1/4}}$ and the definition (3.5) of
$\lambda_{0}$ to get, since $k\geq 2n/\eta$,

$\displaystyle\frac{\rho^{j-v}}{|x_{v}(\omega)|}$
$\displaystyle\leq\rho^{k^{1/4}}\lambda_{0}^{k^{1/4}}\leq\lambda_{0}^{-k^{1/4}}\leq\frac{C}{n^{3}}\leq\frac{C}{(k+n-j)^{3}}\,,\quad
k\leq j\leq k+n-1\,.$

The term $(k+n-j)\lambda_{0}^{-(k-k^{1/4})}$ is similar. Finally,
$|x_{v}(\omega)|\leq n^{-3/\alpha}$ gives

$\sum_{j=k}^{k+n-1}(k+n-j)|x_{v}(\omega)|^{\alpha}\leq\sum_{j=k}^{k+n-1}\frac{k+n-j}{n^{3}}\leq
n\frac{k+n-k}{n^{3}}=\frac{1}{n}\,.$

This proves (3.18), and hence (3.9).

We will next deduce (3.8) from (3.9). Fix $\kappa_{1}>\kappa_{0}$, let
$N_{1}(\kappa_{1})\geq N_{0}$ be given by Lemma 2.3, and let $K\geq N_{1}$ be
such that $k^{\kappa_{1}}\leq\lambda_{0}^{k^{1/4}}$ for all $k\geq K$. Then,
if $v(k)\geq K$ (so that $k\geq K$), we have

$|x_{v}(\tilde{\omega})|>v^{\kappa_{1}}=[k-k^{1/4}]^{\kappa_{1}}>\lambda_{0}^{-k^{1/4}}\,,\quad\forall\tilde{\omega}\in\mathcal{P}_{v}\,.$

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

$\lambda_{0}^{-k^{1/4}}\leq|x_{v}(\omega)|\leq
n^{-3/\alpha}\,,\qquad\forall\omega\in\mathcal{Q}_{v}\,,$

we let $\mathcal{Q}_{v}^{*}$ consist of those intervals in $\mathcal{Q}_{v}$
having a nonempty intersection with $\Omega_{*}^{\varphi}$, setting
$\widetilde{\Omega}_{v}=\cup_{\omega\in\mathcal{Q}_{v}^{*}}\omega$, and we
decompose

$|\Omega^{\varphi}_{*}|\cdot
E\biggl{(}\sum_{j=k}^{k+n-1}\xi_{j}\biggr{)}^{2}=\sum_{\omega\in\mathcal{Q}_{v}^{*}}\int_{\omega}\biggl{(}\sum_{j=k}^{k+n-1}\xi_{k}\biggr{)}^{2}dm-\int_{\widetilde{\Omega}_{v}\setminus\Omega^{\varphi}_{*}}\biggl{(}\sum_{j=k}^{k+n-1}\xi_{k}\biggr{)}^{2}dm\,.$

Then, using (2.9), $\sup_{k}\sup|\xi_{k}|<\infty$, $\kappa_{0}\geq 3/d_{1}$,
and $v(k)\geq k/2\geq n/\eta$,

$\displaystyle\int_{\widetilde{\Omega}_{v}\setminus\Omega^{\varphi}_{*}}\biggl{(}\sum_{j=k}^{k+n-1}\xi_{k}\biggr{)}^{2}dm$
$\displaystyle\leq\int_{\Omega_{v}\setminus\Omega^{\varphi}_{*}}\biggl{(}\sum_{j=k}^{k+n-1}\xi_{k}\biggr{)}^{2}dm$
$\displaystyle\leq Cn^{2}e_{v}\leq Cn^{2}n^{1-d_{1}\kappa_{0}}\leq C\,,$

while (3.9) gives

(3.19)
$\int_{\omega}\biggl{(}\sum_{j=k}^{k+n-1}\xi_{k}\biggr{)}^{2}dm=|\omega|(n+O(1))\,,\qquad\forall\omega\in\mathcal{Q}_{v}^{*}\,.$

Summing (3.19) over $\omega\in\mathcal{Q}^{*}_{v}$, and using again (2.9) to
see

$0\leq\frac{|\widetilde{\Omega}_{v}|}{|\Omega^{\varphi}_{*}|}-1\leq\frac{|\Omega_{v}|}{|\Omega^{\varphi}_{*}|}-1=O(e_{v})=O(e_{n})\,,$

we have established (3.8), ending 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 $j$th block
$\mathbb{I}_{j}$ is $j^{2/3}$, which will give the error exponent $\gamma>2/5$
in our ASIP.313131A block size $\\#\mathbb{I}_{j}=j^{b}$ replaces $3/5$ in
(4.4) by $1/(1+b)$, so that the first constraint becomes
$N^{\gamma}>N^{b/(1+b)}$, see (4.5). Our bounds (4.22)–(4.23) (with Gál–Koksma
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 iid case a block size $j^{1/2}$ gives
$\gamma>1/3$ ([PS, p. 25]), see also the beginning of [Sch, Sec. 6].

###  4.1. Blocks $\mathbb{I}_{M}$. Approximations $\chi_{i}$ and $y_{j}$

Fix $a_{*}$, $\varpi\in(0,1)$, $q$,
$s\in(0,\operatorname{min}\\{\varpi,1/q\\})$, $\rho$, $\theta$, $\lambda_{0}$,
$\eta$, $\alpha$, $\varphi\in C^{\varpi}$,
$\Omega_{*}^{\varphi}=\Omega_{*}\cap[a_{*}-\epsilon_{\varphi},a_{*}+\epsilon_{\varphi}]$
as in the beginning of Section 3. Set

$\mathcal{P}_{*,k}:=\\{\omega\in\mathcal{P}_{k}\mid|\omega\cap\Omega_{*}^{\varphi}|>0\\}\,,\qquad\Omega_{*,k}:=\bigcup_{\omega\in\mathcal{P}_{*,k}}\omega,\qquad
k\geq 1\,.$

Fix $\gamma\in(2/5,1/2)$ and323232See Lemma 4.4 for the condition
$\delta<2(\gamma-2/5)$.
$\delta\in(0,\operatorname{min}\\{1/5,2(\gamma-2/5)\\})$. For $i\geq 1$, we
shall approach
$\xi_{i}\colon[a_{*}-\epsilon_{\varphi},a_{*}+\epsilon_{\varphi}]\to\mathbb{C}$
(see Remark 3.1) by the stepfunction

$\chi_{i}\colon\Omega_{*,r_{i}}\to\mathbb{C}\,,\qquad\chi_{i}=E(\xi_{i}|\mathcal{F}_{r_{i}})\,,\,\,\,\,\mbox{
where }r_{i}=i+[i^{\delta}]\,,$

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 (3.7))

(4.1)
$\chi_{i}|_{\omega}\equiv\frac{\int_{\omega\cap\Omega_{*}^{\varphi}}\xi_{i}\,dm}{|\omega\cap\Omega_{*}^{\varphi}|}\,,\qquad\forall\omega\in\mathcal{P}_{*,r_{i}}\,,\,\forall
i\geq 1\,.$

Thus, $\chi_{i}$ is defined everywhere on $\Omega_{*,r_{i}}$, allowing
pointwise claims about it.

Recalling $e_{\ell}$ from (2.9), we have the following basic lemma:

######  Lemma 4.1.

There exists $C$ such that

(4.2) $|\xi_{i}(a)-\chi_{i}(a)|\leq C\lambda_{CE}^{-\theta
i^{\delta}}\,,\qquad\forall i\geq 1\,,\ \forall a\in\Omega_{*,r_{i}}\,,$

and333333The constant $C$ in (4.3) goes to infinity as $\delta\to 0$, i.e. if
$\gamma\to 2/5$. for all $i\geq 1$, $j\geq 0$ and all $a\in\Omega_{*,r_{i}}$

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

Following [PS, Sec 3.3], [Sch, Sec 6.1], we define inductively consecutive
blocks $\mathbb{I}_{j}$ of integers and associated functions $y_{j}$: Let
$\mathbb{I}_{1}=\\{1\\}$, and let $\mathbb{I}_{j}$ for $j\geq 2$ contain
$[j^{2/3}]$ consecutive integers. The first blocks are below:

$\underbrace{1,}_{\mathbb{I}_{1}}\underbrace{2,}_{\mathbb{I}_{2}}\underbrace{3,4,}_{\mathbb{I}_{3}}\underbrace{5,6,}_{\mathbb{I}_{4}}\underbrace{7,8,}_{\mathbb{I}_{5}}\underbrace{9,10,11,}_{\mathbb{I}_{6}}\underbrace{12,13,14,}_{\mathbb{I}_{7}}\underbrace{15,16,17,18,}_{\mathbb{I}_{8}}\underbrace{19,20,21,22,}_{\mathbb{I}_{9}}\ldots$

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

(4.4) $C^{-1}N^{3/5}\leq M(N)\leq CN^{3/5}\,,\qquad\forall N\geq 1\,.$

By (4.2) in Lemma 4.1, there is $C$ such that, for all $i\geq 1$ and all
$a\in\Omega_{*,r_{i}}$,

(4.5)
$\displaystyle\biggl{|}\sum_{i=1}^{N}\xi_{i}(a)-\sum_{j=1}^{M(N)}\sum_{i\in\mathbb{I}_{j}}\chi_{i}(a)\biggr{|}$
$\displaystyle\leq\sum_{i=1}^{N}|\xi_{i}(a)-\chi_{i}(a)|+C\\#\mathbb{I}_{M}\leq
CN^{2/5}\,,$

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

$y_{j}\colon\Omega_{*,[Cr_{j}^{5/3}]}\to\mathbb{C}\,,\qquad
y_{j}:=\sum_{i\in\mathbb{I}_{j}}\chi_{i}\,,\qquad j\geq 1\,.$

###### Proof of Lemma 4.1.

By (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 (3.3),
and using (3.2) and the $\theta$-Hölder continuity of $a\mapsto\varphi_{a}(u)$
(as for (3.14)), we find $C$ such that for all $i\geq 1$ and
$\omega\in\mathcal{P}_{*,r_{i}}$

$\displaystyle|\xi_{i}(a)-\chi_{i}(a)|=|\xi_{i}(a)-\xi_{i}(a^{\prime})|$
$\displaystyle\leq C(|\omega|^{\theta}+|x_{i}(\omega)|^{\varpi})\leq
C|x_{i}(\omega)|^{\theta}\,,\,\forall a\in\omega\,,$

where we used $\theta\leq\varpi$ and (2.7) in second inequality. This
establishes (4.2), since $|x_{i}(\omega)|\leq C\lambda_{CE}^{-i^{\delta}}$ for
all $\omega\in\mathcal{P}_{r_{i}}$, by (2.29) and (2.5).

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

(4.6) $\displaystyle|\omega\cap\Omega_{*}^{\varphi}|\cdot$
$\displaystyle|E(\xi_{i+j}|\mathcal{F}_{r_{i}})(a)|=\int_{\omega\cap\Omega_{*}^{\varphi}}\xi_{i+j}\,dm$
$\displaystyle=\sum_{\begin{subarray}{c}\omega^{\prime}\in\mathcal{P}_{*,r_{i+j}}\\\
\omega^{\prime}\subset\omega\end{subarray}}|\omega^{\prime}\cap\Omega_{*}^{\varphi}|\cdot\frac{\int_{\omega^{\prime}\cap\Omega_{*}^{\varphi}}\xi_{i+j}\,dm}{|\omega^{\prime}\cap\Omega_{*}^{\varphi}|}$
$\displaystyle=\sum_{\begin{subarray}{c}\omega^{\prime}\in\mathcal{P}_{*,r_{i+j}}\\\
\omega^{\prime}\subset\omega\end{subarray}}|\omega^{\prime}\cap\Omega_{*}^{\varphi}|\cdot\chi_{i+j}|_{\omega^{\prime}}=\sum_{\begin{subarray}{c}\omega^{\prime}\in\mathcal{P}_{*,r_{i+j}}\\\
\omega^{\prime}\subset\omega\end{subarray}}\int_{\omega^{\prime}\cap\Omega_{*}^{\varphi}}\chi_{i+j}\,dm\,.$

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

(4.7)
$k=k(i,j)=\operatorname{max}\biggl{\\{}i+[i^{\delta}]+\eta(j-i^{\delta}),\biggl{\lceil}\frac{i+j}{1+\eta}\biggr{\rceil}\biggr{\\}}$

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

Since $\delta$ is fixed, we may and shall assume that $i$ is large enough such
that $k(i,j)\geq N_{1}$ (with $N_{1}$ from Lemma 2.3) and

(4.8)
$\operatorname{max}\\{\lambda_{0}^{-\eta(j+i)},\rho^{\eta(j-i^{\delta})/(1+\eta)},\rho^{\eta
j/3}\cdot(2j)^{(\kappa_{0}+1)/\delta}\\}\leq e_{[\eta(j-i^{\delta})]}\,.$

Since $k(i,j)\geq r_{i}$, we have, similarly as for (4.6),

$|E(\xi_{i+j}|\mathcal{F}_{r_{i}})(a)|=|E(E(\xi_{i+j}|\mathcal{F}_{k(i,j)})|\mathcal{F}_{r_{i}})(a)|\,,\qquad\forall
a\in\tilde{\omega}\in\mathcal{P}_{*,r_{i}}\,.$

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

(4.9)
$|\tilde{\omega}\cap\Omega_{*}^{\varphi}|\cdot|E(E(\xi_{i+j}|\mathcal{F}_{k(i,j)})|\mathcal{F}_{r_{i}})(a)|=\biggl{|}\sum_{\begin{subarray}{c}\omega\in\mathcal{P}_{*,k(i,j)}\\\
\omega\subset\tilde{\omega}\end{subarray}}\frac{|\omega\cap\Omega^{\varphi}_{*}|}{|\omega\cap\Omega^{\varphi}_{*}|}\int_{\omega\cap\Omega_{*}^{\varphi}}\xi_{i+j}\,dm\biggr{|}\\\
\leq\biggl{|}\sum_{\begin{subarray}{c}\omega\in\mathcal{P}_{*,k(i,j)}\\\
\omega\subset\tilde{\omega}\end{subarray}}\frac{|\omega|}{|\omega|}\int_{\omega}\xi_{i+j}\,dm\biggr{|}+\sup_{a}\|\varphi_{a}\|_{L^{\infty}}\cdot\sum_{\begin{subarray}{c}\omega\in\mathcal{P}_{*,k(i,j)}\\\
\omega\subset\tilde{\omega}\end{subarray}}|\omega\setminus(\omega\cap\Omega^{\varphi}_{*})|\,.$

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

(4.10)
$\begin{cases}\frac{\sum_{\begin{subarray}{c}\omega\in\mathcal{P}_{*,k(i,j)}\\\
\omega\subset\tilde{\omega}\end{subarray}}|\omega\setminus(\omega\cap\Omega_{*})|}{|\tilde{\omega}\cap\Omega_{*}^{\varphi}|}\leq\frac{d_{0}e_{k(i,j)-r_{i}}|\tilde{\omega}|}{(1-d_{0}e_{k(i,j)-r_{i}})|\tilde{\omega}|}\leq
Cd_{0}e_{[\eta(j-i^{\delta})]}\,,\\\
|\tilde{\omega}|/{|\tilde{\omega}\cap\Omega_{*}^{\varphi}|}\leq\frac{|\tilde{\omega}|}{(1-d_{0}e_{k(i,j)-r_{i}})|\tilde{\omega}|}\leq
C\,.\end{cases}$

In view of (2.10), (4.10) and (4.9), it suffices to show

$\frac{1}{|\omega|}\biggl{|}\int_{\omega}\xi_{i+j}\,dm\biggr{|}\leq
C\operatorname{min}(1,e_{[\eta(j-2i^{\delta})]})\,,\qquad\forall\omega\in\mathcal{P}_{*,k(i,j)}\,.$

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

(4.11)
$\frac{1}{|\omega|}\biggl{|}\int_{\omega}\xi_{i+j}(a)\,da\biggr{|}\leq\frac{C}{|x_{k}(\omega)|}\biggl{|}\int_{x_{k}(\omega)}\xi_{i+j}(x_{k}|_{\omega}^{-1}(y))\,dy\biggr{|}\,.$

Then, on the one hand, Lemma 3.3 for $\ell_{0}=1$ gives
$a_{0}\in\omega\cap\Omega^{\varphi}_{*}$ such that

(4.12)
$\displaystyle\frac{1}{|x_{k}(\omega)|}\biggl{|}\int_{x_{k}(\omega)}(\xi_{i+j}(x_{k}|_{\omega}^{-1}(y))$
$\displaystyle-\varphi_{a_{0}}(T_{a_{0}}^{i+j-k}(y)))\,dy\biggr{|}$
$\displaystyle\leq C\lambda_{0}^{-k(i,j)}\leq
C\lambda_{0}^{-(i+j)/(1+\eta)}\,.$

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
$\int\varphi_{a_{0}}d\mu_{a_{0}}=0$, gives343434A factor
$|x_{k}(\omega)|^{-1}$ was omitted when applying [Sch, Prop. 4.3] on p. 400 of
[Sch]: We fix this by using our polynomial lower bound on $|x_{k}(\omega)|$
(considering two different values of $\delta$ should work for [Sch]).

$\displaystyle\frac{1}{|x_{k}(\omega)|}\biggl{|}$
$\displaystyle\int_{x_{k}(\omega)}\varphi_{a_{0}}(T_{a_{0}}^{i+j-k}(y))\,dy\biggr{|}\leq
C\cdot{k(i,j)^{\kappa_{1}}}\rho^{i+j-k(i,j)}$ (4.13) $\displaystyle\,\leq
C\cdot(i+j)^{\kappa_{1}}\rho^{\eta(j-i^{\delta})/(1+\eta)}\leq
C\cdot(2j)^{\kappa_{1}/\delta}\rho^{\eta j/(2+2\eta)}\,.$

(We used $|x_{k}(\omega)|>Ck^{-\kappa_{1}}$ from Lemma 2.3.) Putting together
(4.11), (4.12), (4.13) and (4.8), we conclude the proof of (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

(4.14) $\biggl{|}N-\sum_{j=1}^{M(N)}y_{j}^{2}(a)\biggr{|}\leq
C(a)N^{2\gamma}\,,\qquad\forall N\geq 1\,.$

The proof of Lemma 4.2 (which uses Proposition 3.2 and (4.2), but not (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 $z_{j}$, $j\geq 1$, be zero-mean random variables. Assume there exist
$p\geq 1$ and $C<\infty$ with

$E\biggl{(}\sum_{j=m+1}^{m+n}z_{j}\biggr{)}^{2}\leq
C((m+n)^{p}-m^{p})\,,\quad\forall\ m\geq 0\ \text{and}\ n\geq 1\,.$

Then for all $\iota>0$, we have $\frac{1}{n^{p/2+\iota}}\sum_{j=1}^{n}z_{j}\to
0$ 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 $|y_{j}+w_{j}|\leq
Cj^{2/3}$, the bound (4.2) gives $C$ such that $y_{j}^{2}-w_{j}^{2}\leq
Cj^{2/3}\lambda_{CE}^{-\theta j^{\delta}}$ for all $j\geq 1$ 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 (4.14) with $y_{j}$ replaced by $w_{j}$.

By (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
$\bigl{|}\sum_{j=1}^{M(N)}E(w_{j}^{2})-N\bigr{|}\leq CM(N)$. Therefore,

(4.15) $\Biggl{|}N-\sum_{j=1}^{M(N)}w_{j}^{2}\Biggr{|}\leq
CM(N)+\Biggl{|}\sum_{j=1}^{M(N)}w_{j}^{2}-E(w_{j}^{2})\Biggr{|}\,.$

Assume there exists $C$ such that

(4.16) $E\biggl{(}\sum_{j=m+1}^{m+n}w_{j}^{2}-E(w_{j}^{2})\biggr{)}^{2}\leq
C((m+n)^{8/3}-m^{8/3})\,,\quad\forall m\geq 0,\ n\geq 1\,.$

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

$\sum_{j=1}^{M(N)}w_{j}^{2}-E(w_{j}^{2})=o(M^{\frac{4}{3}+\iota})\,,\qquad\mbox{almost
surely}\,.$

Hence, (4.15) gives $\bigl{|}N-\sum_{j=1}^{M(N)}w_{j}^{2}(a)\bigr{|}\leq
C(a)N^{4/5+3\iota/5}\leq C(a)N^{2\gamma}$, almost surely (recall $M(N)\sim
N^{3/5}$ by (4.4)). It remains to prove (4.16).

By Jensen’s inequality we have $(E(w_{j}^{2}))^{2}\leq E(w_{j}^{4})$ and
therefore

(4.17) $E\biggl{(}\sum_{j=m+1}^{m+n}w_{j}^{2}-E(w_{j}^{2})\biggr{)}^{2}\\\
\leq
2\sum_{j=m+1}^{m+n}\biggl{(}E(w_{j}^{4})+\sum_{k=j+1}^{m+n}|E(w_{j}^{2}w_{k}^{2})-E(w_{j}^{2})E(w_{k}^{2})|\biggr{)}\,.$

We consider first $E(w_{j}^{4})$. Fix $\upsilon\in(0,1/6)$ and, for $j\geq 1$,
let

$S_{j}=\\{\,\vec{v}\in\mathbb{I}_{j}^{4}\mid v_{1}\leq v_{2}\leq v_{3}\leq
v_{4}\mbox{ and }\operatorname{min}\\{v_{2}-v_{1},v_{4}-v_{3}\\}\geq
j^{\upsilon}\,\\}.$

Then, since $\\#(\mathbb{I}_{j}^{4}\setminus
S_{j})\leq(j^{2/3+\upsilon})^{2}=j^{4/3+2\upsilon}$, we find

$\displaystyle\int\limits_{\Omega_{*}^{\varphi}}w_{j}(a)^{4}\,da$
$\displaystyle=\sum_{\vec{v}\in\mathbb{I}_{j}}\biggl{|}\int_{\Omega_{*}^{\varphi}}\prod_{\ell=1}^{4}\xi_{v_{\ell}}(a)\,da\biggr{|}\leq
C\sum_{\begin{subarray}{c}\vec{v}\in\mathbb{I}_{j}\\\ v_{1}\leq\ldots\leq
v_{4}\end{subarray}}\biggl{|}\int_{\Omega_{*}^{\varphi}}\prod_{\ell=1}^{4}\xi_{v_{\ell}}(a)\,da\biggr{|}$
(4.18) $\displaystyle\leq C\sum_{\vec{v}\in
S_{j}}\biggl{|}\int_{\Omega_{*}^{\varphi}}\prod_{\ell=1}^{4}\xi_{v_{\ell}}(a)\,da\biggr{|}+Cj^{4/3+2\upsilon}\,.$

Next we estimate the double sum over $j$ and $k$ in (4.17). 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
$\omega\cap\Omega_{*}^{\varphi}\neq\emptyset$, the change of variable in
equation (3.11), together with an easy variant of Lemma 3.3 deduced from
(3.12), give $a_{0}\in\omega\cap\Omega_{*}^{\varphi}$ such that

$\displaystyle\frac{1}{|\omega|}\biggl{|}\int_{\omega}\prod_{\ell=1}^{4}\xi_{\ell}(a)\,da\biggr{|}\leq\frac{C}{|x_{v_{3}}(\omega)|}\biggl{|}\int_{x_{v_{3}}(\omega)}\biggl{(}\prod_{\ell=1}^{3}\xi_{\ell}(x_{v_{3}}|_{\omega}^{-1}(y))\biggr{)}$
$\displaystyle\varphi_{a_{0}}(T_{a_{0}}^{v_{4}-v_{3}}(y))\,dy\biggr{|}$
$\displaystyle+C\lambda_{0}^{-v_{3}}\,.$

For $y\in x_{v_{3}}(\omega)$, setting $a=x_{v_{3}}|_{\omega}^{-1}(y)$, and
recalling Remark 3.1, we find

$\displaystyle|\xi_{v_{\ell}}(x_{v_{3}}|_{\omega}^{-1}(y))$
$\displaystyle-\varphi_{a_{0}}(x_{v_{\ell}}\circ
x_{v_{3}}|_{\omega}^{-1}(y))|$ $\displaystyle=|\varphi_{a}(x_{v_{\ell}}\circ
x_{v_{3}}|_{\omega}^{-1}(y))-\varphi_{a_{0}}(x_{v_{\ell}}\circ
x_{v_{3}}|_{\omega}^{-1}(y))|\leq C|\omega|^{\theta}\,,$

for $\ell=1,2,3$. Thus, (2.9) and (2.7) imply (using
$\sup_{k}\|\xi_{k}\|_{L^{\infty}}<\infty$)

(4.19)
$\biggl{|}\int_{\Omega^{\varphi}_{*}}\prod_{\ell=1}^{4}\xi_{v_{\ell}}(a)\,da\biggr{|}\leq
Ce_{v_{3}}+\sum_{\omega\in\mathcal{P}_{*,v_{3}}}|\omega|\biggl{[}\frac{C\lambda_{CE}^{-v_{3}\theta}}{|x_{v_{3}}(\omega)|}+C\lambda_{0}^{-v_{3}}\biggr{]}\\\
+\sum_{\omega\in\mathcal{P}_{*,v_{3}}}|\omega|\frac{C}{|x_{v_{3}}(\omega)|}\Bigl{|}\int_{x_{v_{3}}(\omega)}\Bigl{(}\prod_{\ell=1}^{3}\varphi_{a_{0}}(x_{v_{\ell}}\circ(x_{v_{3}}|_{\omega}^{-1})(y))\Bigr{)}\varphi_{a_{0}}(T_{a_{0}}^{v_{4}-v_{3}}(y))\,dy\Bigr{|}\,.$

By (2.9) and (2.29), we have, for $\ell=1,2,3$, and for each
$\omega\in\mathcal{P}_{*,v_{3}}$,

$|\partial_{y}(x_{v_{\ell}}\circ(x_{v_{3}}|_{\omega}^{-1}))(y)|\leq
C\lambda^{-(v_{3}-v_{\ell})}_{CE}\,,\quad\forall y\in x_{v_{3}}(\omega)\,.$

Therefore, we find $C$ such that for each $v_{3}$ and
$\omega\in\mathcal{P}_{*,v_{3}}$,

$\|1_{x_{v_{3}}(\omega)}\cdot\prod_{\ell=1}^{3}(\varphi_{a_{0}}\circ
x_{v_{\ell}}\circ x_{v_{3}}|_{\omega}^{-1})\|_{H^{s}_{q}}\leq
C\|\varphi_{a_{0}}\|^{2}_{C^{\varpi}}\|\varphi_{a_{0}}\|_{H^{s}_{q}}\,.$

Indeed, composition with a $C^{2}$ contraction is a bounded operator on
$C^{\varpi}$ or $H^{s}_{q}(I)$, 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), 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 (3.2) and
$\int\varphi_{a_{0}}d\mu_{a_{0}}=0$), we have

$\frac{\bigl{|}\int_{x_{v_{3}}(\omega)}\bigl{(}\prod_{\ell=1}^{3}\varphi_{a_{0}}(x_{v_{\ell}}\circ
x_{v_{3}}|_{\omega}^{-1})\bigr{)}\varphi_{a_{0}}(T_{a_{0}}^{v_{4}-v_{3}})\,dy\bigr{|}}{|x_{v_{3}}(\omega)|}\leq
C\frac{\rho^{v_{4}-v_{3}}}{|x_{v_{3}}(\omega)|}\leq
Cv_{3}^{\kappa_{1}}\rho^{j^{\upsilon}}\,.$

(We used Lemma 2.3). Since $v_{3}\in\mathbb{I}_{j}$, we have $v_{3}\leq
Cj^{5/3}$, and

$\biggl{|}\int_{\Omega^{\varphi}_{*}}\prod_{\ell=1}^{4}\xi_{v_{\ell}}\,da\biggr{|}\leq
C(e_{[Cj^{5/3}]}+j^{5\kappa_{1}/3}\rho^{j^{\upsilon}}+j^{5\kappa_{1}/3}\lambda_{0}^{j^{5/3}})\leq
Ce_{[Cj^{5/3}]}\,,$

for all $\vec{v}\in S_{j}$ with $v_{4}-v_{3}\geq j^{\upsilon}$ (if $j$ is
large enough).

Let now $\vec{v}\in S_{j}$ with $v_{2}-v_{1}\geq j^{\upsilon}$. Then applying
directly Lemma 3.3 with $\ell_{0}=4$, a similar reasoning gives
$|\int_{\Omega_{*}^{\varphi}}\prod_{\ell=1}^{4}\xi_{v_{\ell}}\,da|\leq
Ce_{[Cj^{5/3}]}$.

Finally, since $\\#S_{j}\leq\\#\mathbb{I}_{j}^{4}\leq j^{8/3}$ and $e_{j}\leq
j^{-d_{1}\kappa_{0}+1}$ with $d_{1}\kappa_{0}\geq 3>9/5$, the bound (4.18)
gives $C$ such that353535For the purposes of the present lemma, a version of
(4.20) with $Cj^{5/3}$ in the right-hand side would suffice. The stronger
statement is needed for (4.28).

(4.20) $E(w_{j}^{4})\leq C(j^{8/3}e_{[Cj^{5/3}]}+j^{4/3+2\upsilon})\leq
Cj^{4/3+2\upsilon}\,,\qquad\forall j\geq 1\,.$

We next bound $|E(w_{j}^{2}w_{k}^{2})-E(w_{j}^{2})E(w_{k}^{2})|$ for $k\geq
j+1$. If $k=j+1$, by Cauchy’s inequality and (4.20),

$E(w_{j}^{2}w_{j+1}^{2})\leq\sqrt{E(w_{j}^{4})E(w_{j+1}^{4})}\leq Cj^{5/3}\,.$

By Proposition 3.2, we have $E(w_{j}^{2})E(w_{j+1}^{2})\leq Cj^{4/3}$. Hence

(4.21) $|E(w_{j}^{2}w_{j+1}^{2})-E(w_{j}^{2})E(w_{j+1}^{2})|\leq Cj^{5/3}\,.$

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 $k_{0}\geq k^{5/3}/C$
be the smallest number in $\mathbb{I}_{k}$. Then, for large enough $j$, using
that $x\mapsto x-x^{1/4}$ is increasing for large $x$, we find

$k_{0}-k_{0}^{1/4}\geq j_{1}+2j_{1}^{2/3}-(j_{1}+2j_{1}^{2/3})^{1/4}\geq
j_{1}+2j_{1}^{2/3}-2j_{1}^{1/4}\geq j_{1}+j_{1}^{1/4}\,.$

Thus, $y_{j}$ is constant on elements of $\mathcal{P}_{v}$ for
$v=[k_{0}-k_{0}^{1/4}]$ (recall $\delta<\frac{1}{4}$).

Lemma 2.3 gives $|x_{v}(\omega)|\geq\lambda_{0}^{-{k_{0}}^{1/4}}$ if
$\omega\in\mathcal{P}_{v}$. Thus, there exists a refinement $\mathcal{Q}_{v}$
of $\mathcal{P}_{v}$ such that (using
$\lambda_{0}^{-{k_{0}}^{1/4}}\leq\lambda_{0}^{-{{k}^{5/12}/C}}$),

$\lambda_{0}^{-{k_{0}}^{1/4}}\leq|x_{v}(\omega)|\leq\\#\mathbb{I}_{k}^{-3/\alpha}=[k^{2/3}]^{-3/\alpha}\,,\quad\forall\omega\in\mathcal{Q}_{v}\,.$

Therefore, for large enough $k$, the local bound (3.9) in Proposition 3.2
gives

$\bigl{|}E\bigl{(}w_{k}^{2}\mid\omega\bigr{)}-\\#\mathbb{I}_{k}\bigl{|}\leq
C\,,\qquad\forall\omega\in\mathcal{Q}_{v}\,.$

Thus, using that $y_{j}$ is constant on each $\omega\in\mathcal{Q}_{v}$ (since
$\mathcal{Q}_{v}$ refines $\mathcal{P}_{v}$),

$\displaystyle E(y_{j}^{2}w_{k}^{2})$
$\displaystyle=\sum_{\omega\in\mathcal{Q}_{v}}|\omega|\cdot
y_{j}^{2}|_{\omega}\cdot
E\bigl{(}w_{k}^{2}\mid\omega\bigr{)}\in[E(y_{j}^{2})(\\#\mathbb{I}_{k}-C),E(y_{j}^{2})(\\#\mathbb{I}_{k}+C)]\,.$

That is, $|E(y_{j}^{2}w_{k}^{2})-\\#\mathbb{I}_{k}E(y_{j}^{2})|\leq
CE(y_{j}^{2})$. Next, the global estimate (3.8) in Proposition 3.2 gives
$|E(y_{j}^{2})E(w_{k}^{2})-\\#\mathbb{I}_{k}E(y_{j}^{2})|\leq CE(y_{j}^{2})$.
Therefore363636The expression $\\#\mathbb{I}_{j}=j^{b}=j^{2/3}$ in the right-
hand side already leads to $\gamma>2/5$. See Footnote 31.

$|E(y_{j}^{2}w_{k}^{2})-E(y_{j}^{2})E(w_{k}^{2})|\leq 2CE(y_{j}^{2})\leq
C\\#\mathbb{I}_{j}\,.$

Hence, for large enough $j$ and all $k\geq j+2$, since $\sup|w_{j}+y_{j}|\leq
C\\#\mathbb{I}_{j}$,

$\displaystyle|E(w_{j}^{2}w_{k}^{2})-E(w_{j}^{2})E(w_{k}^{2})|\leq|E(y_{j}^{2}w_{k}^{2})-E(y_{j}^{2})E(w_{k}^{2})|$
$\displaystyle\qquad\qquad+|E(y_{j}^{2}w_{k}^{2})-E(w_{j}^{2}w_{k}^{2})|+|E(w_{j}^{2})E(w_{k}^{2})-E(y_{j}^{2})E(w_{k}^{2})|$
$\displaystyle\leq
C\\#\mathbb{I}_{j}+CE(w_{k}^{2})\sup|w_{j}-y_{j}|\cdot\sup|w_{j}+y_{j}|$
(4.22) $\displaystyle\leq Cj^{2/3}+Ck^{2/3}j^{2/3}\lambda_{CE}^{-\theta
j^{5\delta/3}}\,.$

(We used (4.2) to get $\sup|w_{j}-y_{j}|\leq
C\\#\mathbb{I}_{j}\lambda_{CE}^{-\theta j^{5\delta/3}}$.)

Finally, we plug (4.22), (4.21), (4.20) into (4.17), and get, since
$2\upsilon<1/3$,

$\displaystyle E\biggl{(}$
$\displaystyle\sum_{j=m+1}^{m+n}w_{j}^{2}-E(w_{j}^{2})\biggr{)}^{2}$
$\displaystyle\leq
C\sum_{k=m+3}^{m+n}k^{2/3}\sum_{j=m+1}^{\infty}j^{2/3}\lambda_{CE}^{-\theta
j^{5\delta/3}}+C\sum_{j=m+1}^{m+n}\biggl{(}j^{5/3}+\sum_{k=j+2}^{m+n}j^{2/3}\biggr{)}$
(4.23) $\displaystyle\leq
C\Biggl{(}(m+n)^{5/3}-m^{5/3}+\sum_{j=m+1}^{m+n}\biggl{(}j^{5/3}+(m+n-j)j^{2/3}\biggr{)}\Biggr{)}\,.$

This proves (4.16). ∎

###  4.3. Martingale Differences $Y_{j}$. Skorokhod’s Representation Theorem

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

(4.24) $\displaystyle u_{j}$ $\displaystyle=\sum_{k\geq
0}E(y_{j+k}\mid\mathcal{L}_{j-1})\,,\qquad Y_{j}=y_{j}+u_{j+1}-u_{j}\,,\qquad
j\geq 2\,.$

Then $\\{Y_{j},\mathcal{L}_{j}\\}$ is a martingale difference sequence. Using
(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

(4.25) $\Biggl{|}N-\sum_{j=1}^{M(N)}Y_{j}^{2}(a)\Biggr{|}\leq
C(a)N^{2\gamma}\,,\qquad\forall N\geq 1\,,$

and

(4.26)
$\Biggl{|}\sum_{j=1}^{M(N)}E(Y_{j}^{2}\mid\mathcal{L}_{j-1})-Y_{j}^{2}(a)\Biggr{|}\leq
C(a)N^{2\gamma}\,,\qquad\forall N\geq 1\,.$

###### 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
(4.4). Then

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

Since $\sum_{j=1}^{\infty}e_{j}<\infty$, the bound (4.3) in Lemma 4.1 gives

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

Put $v_{j}=u_{j}-u_{j+1}$, so that
$Y_{j}^{2}=y_{j}^{2}-2y_{j}v_{j}+v_{j}^{2}$.

We claim that (4.25) follows if for a.e. $a\in\Omega^{\varphi}_{*}$, there
exists $C$ such that $\sum_{j=1}^{M(N)}v_{j}^{2}\leq CN^{4\gamma-1}$. Indeed,
since $\gamma<1/2$, Lemma 4.2 and Cauchy’s inequality then give (using
$\sum_{j=1}^{M(N)}y_{j}^{2}\leq CN$)

$\displaystyle\Biggl{|}N-\sum_{j=1}^{M(N)}Y_{j}^{2}\Biggr{|}$
$\displaystyle=\Biggl{|}N-\sum_{j=1}^{M(N)}(y_{j}^{2}-2y_{j}v_{j}+v_{j}^{2})\Biggr{|}$
$\displaystyle\leq\Biggl{|}N-\sum_{j=1}^{M(N)}y_{j}^{2}\Biggr{|}+\sum_{j=1}^{M(N)}v_{j}^{2}+2\sqrt{\sum_{j=1}^{M(N)}y_{j}^{2}\sum_{j=1}^{M(N)}v_{j}^{2}}$
$\displaystyle\leq C(a)N^{2\gamma}+CN^{2\gamma}+C\sqrt{NN^{4\gamma-1}}\leq
C(a)N^{2\gamma}\,.$

But since we have $v_{j}^{2}\leq Cj^{10\delta/3}$ (by (4.27)), we find, using
$\delta<2(\gamma-2/5)$,

$\sum_{j=1}^{M(N)}v_{j}^{2}\leq CM^{1+10\delta/3}\leq N^{3/5+2\delta}\leq
CN^{4\gamma-1}\,.$

It remains to prove (4.26). 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

$E(R_{j}^{2})\leq\Bigl{(}\sqrt{E(Y_{j}^{4})}+\sqrt{E(E(Y_{j}^{2}\mid\mathcal{L}_{j-1})^{2})}\Bigr{)}^{2}\leq\Bigl{(}2\sqrt{E(Y_{j}^{4})}\Bigr{)}^{2}=4E(Y_{j}^{4})\,.$

Since $Y_{j}=y_{j}-v_{j}$, we have, again by Minkowski’s inequality,

$\displaystyle E(R_{j}^{2})$ $\displaystyle\leq 4E(Y_{j}^{4})\leq
4\Bigl{(}(E(y_{j}^{4}))^{\frac{1}{4}}+(E(v_{j}^{4}))^{\frac{1}{4}}\Bigr{)}^{4}\leq
C(E(y_{j}^{4})+E(v_{j}^{4}))$ $\displaystyle\leq
C(E(w_{j}^{4})+E(|w_{j}^{4}-y_{j}^{4}|)+E(v_{j}^{4}))\,.$

Since $w_{j}^{4}-y_{j}^{4}=(w_{j}^{2}+y_{j}^{2})(w_{j}+y_{j})(w_{j}-y_{j})$,
we get from (4.2) that $E(|w_{j}^{4}-y_{j}^{4}|)$ is uniformly bounded. By
(4.27), we have $|u_{j}|\leq Cj^{5\delta/3}$. Hence,
$|v_{j}|\leq|u_{j}|+|u_{j-1}|\leq Cj^{5\delta/3}$, and $E(v_{j}^{4})\leq
Cj^{20\delta/3}\leq Cj^{4/3}$, since $\delta<1/5$. For arbitrary $\iota>0$ the
bound (4.20), gives $C$ such that $E(w_{j}^{4})\leq Cj^{4/3+\iota}$. Thus

(4.28) $\displaystyle\sum_{j\geq
1}\frac{E(R_{j}^{2})}{j^{7/3+\iota}}<\infty\,,$

and a martingale result (see [Ch]) implies that $\sum_{j\geq
1}R_{j}/j^{7/6+\iota}$ converges almost surely. For $m_{*}$-a.e.
$a\in\Omega_{*}^{\varphi}$, Kronecker’s Lemma gives $C(a)$ with

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

using (4.4) in the last inequality. Since $21/30<2\gamma$ this establishes
(4.26). ∎

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 nonnegative 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 $j\geq 1$, be the
$\sigma$-algebra generated by

$\\{W(t)\mid 0\leq t\leq\tau_{j}\\}\,,\,\,\mbox{ where
}\tau_{j}:=\sum_{k=1}^{j}T_{k}\,,$

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

$E(T_{j}\mid\mathcal{G}_{j-1})=E\bigl{(}\bigl{(}W(\tau_{j})-W(\tau_{j-1})\bigr{)}^{2}\mid\mathcal{G}_{j-1}\bigr{)}\,,\quad\forall
j\geq 2\,,\quad\mbox{almost surely.}$

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

(4.29)
$E(T_{j}\mid\mathcal{G}_{j-1})=E(W(T_{j})^{2}\mid\mathcal{G}_{j-1})\,,\qquad\forall
j\geq 1\,,$

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

(4.30) $\Biggl{|}N-\sum_{j=1}^{M(N)}T_{j}\Biggr{|}\leq
C(a)N^{2\gamma}\,,\qquad\forall N\geq 1\,.$

###### Proof.

To start, apply Theorem 4.5 to the martingale difference sequence $Y_{j}$ from
(4.24), with $\mathcal{L}_{j}$ generated by $\\{y_{\ell}\\}_{1\leq\ell\leq
j}$. 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
(4.29), we have, almost surely,

$\displaystyle
N-\sum_{j=1}^{M(N)}T_{j}=\bigl{[}N-\sum_{j=1}^{M}\tilde{Y}_{j}^{2}\bigr{]}$
$\displaystyle+\sum_{j=1}^{M}\bigl{[}\tilde{Y_{j}}^{2}-E(\tilde{Y}_{j}^{2}\mid\mathcal{G}_{j-1})\bigr{]}$
$\displaystyle+\sum_{j=1}^{M}\bigl{[}E(T_{j}\mid\mathcal{G}_{j-1})-T_{j}\bigr{]}\,,\,\,\forall
N\geq 1\,.$

Then, since $Y_{j}$ and $\tilde{Y}_{j}$ have the same distribution, the bound
(4.25) in Lemma 4.4 gives $C(a)$ such that, for all $N\geq 1$, 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 (4.26). Since
conditional expectations can be expressed in terms of distributions, (4.26) 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 (4.29).
As in the proof of (4.26), we can estimate $E(R_{j}^{2})\leq
4E(W(T_{j})^{4})$, and thus there exists $C(a)$ such that, for all $N\geq 1$,
we have $\sum_{j=1}^{M(N)}R_{j}\leq CN^{21/30+\iota}\leq C(a)N^{2\gamma}$
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,

$\Biggl{|}\sum_{j=1}^{M(N)}Y_{j}-W(N)\Biggr{|}=O(N^{\gamma})\,.$

Then, using (4.27) and (4.4), we find

(4.31)
$\Biggl{|}\sum_{j=1}^{M(N)}y_{j}-Y_{j}\Biggr{|}=\Biggl{|}\sum_{j=1}^{M(N)}(u_{j+1}-u_{j})\Biggr{|}=|u_{M(N)+1}-u_{1}|\leq
CN^{\delta}\,.$

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

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