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

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

Proof of Proposition 2.2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

For small $\epsilon>0$ (to be chosen depending on $r_0$ ), the sequence $ \mathcal {P}_j$ can now be defined inductively: start with the single interval $ \mathcal {P}_0= \mathcal {P}_1=\cdots = \mathcal {P}_{N_0} =\{\omega _0 \}$ for $\epsilon $ small enough such that $\nu _1( \omega _0)\ge N_0$ (note that $\nu _1(\omega _0)$ increases if $r_0$ increases or $\epsilon $ decreases). (We refer throughout to [Reference de Melo and van StrienDMS, §V.6]. The original ideas and key estimates appeared previously in the work of Benedicks and Carleson [Reference Benedicks and CarlesonBC1, Reference Benedicks and CarlesonBC2]. The original construction in [Reference Benedicks and CarlesonBC1, Reference Benedicks and CarlesonBC2] is for $a_{*}=4$ , see [Reference MoreiraMo] for a self-contained account. It extends to Misiurewicz parameters: for CE parameters, the condition in [Reference de Melo and van StrienDMS, Theorem 6.1] is equivalent to equation (1.4), taking large enough k in the last line of [Reference de Melo and van StrienDMS, p. 406, Step 2].)

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

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

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

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

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

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

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

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

It is well known (the original construction in [Reference Benedicks and CarlesonBC1, Reference Benedicks and CarlesonBC2] is for $a_{*}=4$ , see [Reference MoreiraMo] for a self-contained account. It extends to Misiurewicz parameters: for CE parameters, the condition in [Reference de Melo and van StrienDMS, Theorem 6.1] is equivalent to equation (1.4), taking large enough k in the last line of [Reference de Melo and van StrienDMS, p. 406, Step 2]) [Reference Benedicks and CarlesonBC1, Reference Benedicks and CarlesonBC2, Reference de Melo and van StrienDMS] that, if we replace the condition in equation (2.11) (used to discard intervals) by the exponential condition

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

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

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

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

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

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

(2.15)

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

The key distortion bound [Reference de Melo and van StrienDMS, Proposition V.6.3] gives C such that

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

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

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

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

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

To show that $a_{*}$ is a Lebesgue density point of $\Omega _{*}$ , we may follow [Reference de Melo and van StrienDMS, Step 7 of the proof of Theorem V.6.1], replacing $C e^{-i C_0}$ there by $C^{\prime } i^{-\kappa _0}$ . (We mention a typo there: although the constant $C=C(\epsilon )$ in the unnumbered equation on [Reference de Melo and van StrienDMS, 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 [Reference de Melo and van StrienDMS, Lemma V.6.5].)

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

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

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

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

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

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

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

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

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

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

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

To show equation (2.20), we proceed in three steps, performing the necessary changes in the proof in [Reference de Melo and van StrienDMS, §V.6]. Recall equation (2.12).

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

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

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

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

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

(2.23)

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

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

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

Third, [Reference de Melo and van StrienDMS, Lemmas V.6.7–6.9] are unchanged, establishing equation (2.20).

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

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

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

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

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

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

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

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

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

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

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

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

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

(2.27)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Summarizing,

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

(2.34)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Proof. In the Misiurewicz case, the first claim is [Reference SedroSe, Theorem 10], using Ruelle’s [Reference RuelleRu, Theorem 9, Remark 16.a] decomposition of $h_a$ into the sum of a $C^1$ function and an exponentially decaying sum of ‘spikes’ $x\mapsto |x-c_k(a)|^{-1/2}$ and square root singularities $x\mapsto |x-c_k(a)|^{1/2}$ . For a general $a \in \Omega _\mathrm{BC}$ , set $ T^{-k}_{a,\varsigma }:=(T_a^k|_{U_{k,a,\varsigma }})^{-1}$ for $k\ge 1$ and $\varsigma \in \pm $ , where $U_{k,a,\varsigma }$ is the monotonicity interval of $T_a^k$ containing c, located to the right of c for $\varsigma =+$ , to the left of c for $\varsigma =-$ . Then, since we assumed $\unicode{x3bb} _\mathrm{CE}>e^{ 14 \alpha _\mathrm{BC}}$ in the proof of Proposition 2.2, use [Reference Baladi and SmaniaBS1, Proposition 2.7] that there exist a $C^1$ function $\psi _a \colon I \to {\mathbb R}_+$ and $C^\infty $ functions $\Xi ^k_{a,\pm } \colon [0,1] \to [0,1]$ supported in a neighbourhood of $c_k(a)$ in $T^k_a(U_{k,a,\pm })$ , such that

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

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

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

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

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

Let then

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

(for a suitable $\unicode{x3bb}>1$ ) be the projection from the tower with polynomial recurrence used in [Reference Baladi, Benedicks and SchnellmannBBS], and let $\hat {\mathcal {L}}_{a}$ be the lift $\mathcal {L}_a \Pi _a=\Pi _a \hat {\mathcal {L}}_a$ of the transfer operator $\mathcal {L}_a \varphi (x)=\sum _{T_a(y)=x} \varphi (y)/|T^{\prime }_a(y)|$ . Then ([Reference Baladi, Benedicks and SchnellmannBBS, (66)] gives uniform Lasota–Yorke estimates, [Reference Baladi, Benedicks and SchnellmannBBS, Lemmas 3.8, 4.5 and 4.6, Proposition 4.1] give the weak norm bounds needed by Keller and Liverani [Reference Keller and LiveraniKL]), 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 [Reference Baladi, Benedicks and SchnellmannBBS],

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

where $\hat h_a$ is the fixed point (the fixed point property determines $\hat h_a$ by its value on the level zero of the tower) of $\hat {\mathcal {L}}_a$ on $\mathcal {B}_a^{W^1_1}$ normalized by $\int \Pi _a \hat h_a \, dm = 1$ , while $\hat \nu $ , the non-negative measure whose density with respect to Lebesgue in the level j of the tower is $\unicode{x3bb} ^j$ , is the fixed point of the dual of $\hat {\mathcal {L}}_a$ (see [Reference Baladi and SmaniaBS1, equation (85)], note that $\nu (\hat h_a)=1$ is automatic). Since $\Pi _a\hat h_a=h_a$ and the $W^1_1$ norm dominates any $H^s_q$ norm on I if $s\in [0,1)$ and ${1< q<1/s}$ (by the Sobolev embedding, more precisely [Reference Runst and SickelRS, Ch. 2], the bounded inclusions $ W^1_1 \subset W^\sigma _1=F^{\sigma }_{1,1}\subset F^{\sigma }_{1,2}\subset F^{s}_{q,2}=H^s_q$ , if $\sigma =1+s-1/q\in (0,1)$ and $q\in (1,\infty )$ ), the decomposition in equation (2.40) combined with the uniform bound in equation (2.41) (for $\hat \varphi $ vanishing on all levels $\geq 1$ and constant on level zero of the tower, with $\hat \nu (\varphi )=1$ ) gives the second claim of the sublemma, using again [Reference SedroSe, Lemmas 11 and 12].

Proof of Proposition 2.5

Recall from the proof of Proposition 2.2 that we have ${\unicode{x3bb} _\mathrm{CE}>e^{14 \alpha _\mathrm{BC}}}$ . By mollification, it is enough to prove both bounds for $C^1$ functions $\varphi $ . It is in fact enough to show the first bound for $\varphi \in C^1$ as follows. Indeed, again by mollification (see e.g. the proof of [Reference SedroSe, 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 [Reference Baladi, Benedicks and SchnellmannBBS, Corollary 1.6] by $\Omega _{*}(a_{*}, \kappa _0)$ . The first bound for Lipschitz continuous $\varphi $ thus follows from the second assertion of [Reference Baladi, Benedicks and SchnellmannBBS, Corollary 1.6], since $\Omega _{*}\subset [a_{\mathrm {mix}}, 4)$ . Indeed, note first that a is topologically mixing if and only if its renormalization period $P_{a}$ is equal to one. Second, observe that the constant $C_{\varphi , \psi }$ in the second claim of [Reference Baladi, Benedicks and SchnellmannBBS, Corollary 1.6] can be replaced by $C \|\varphi \|_\varpi \|\psi \|_{L^1(d\mu _a)}$ for a constant C uniform in a in view of [Reference Baladi, Benedicks and SchnellmannBBS, Lemmas 4.5 and 4.6] and the principle of uniform boundedness. More precisely, using the notation from the proof of Sublemma 2.7, we have

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

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

Proof of Proposition 2.6

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

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

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

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

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

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

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

Next, equation (2.2) gives that

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

Therefore, we find

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

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

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

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

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

Proof of Lemma 4.1

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

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

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

(4.6)

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

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

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

(4.7)

(see e.g. [Reference Baladi and SmaniaBS1, Proposition 3.7] in the exponentially recurrent case). If $a\notin \Omega _{*}$ , we may use bounded distortion in equation (2.31) ( $\alpha =0$ suffices here) since $|\omega \cap \Omega _{*}|>0$ .

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

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

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

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

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

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

(4.10)

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

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

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

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

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

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

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

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

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

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

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

(4.14)

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

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

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

Proof of Lemma 4.2

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

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

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

Assume there exists C such that

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

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

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

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

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

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

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

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

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

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

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

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

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

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

(4.21)

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

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

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

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

Thus, if $a\in \Omega _{*}$ , standard arguments (see e.g. [Reference Baladi and SmaniaBS1, Proposition 3.7], using our polynomial recurrence in equation (2.8)) give the claim. Otherwise, since $|\omega \cap \Omega _{*}|>0$ , we may use equation (2.31) as for equation (4.6).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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