####  7.8.5 Deviations of the conjugation I: proof of the bounds (2.42a)

We can now come back to $h(\varphi,\psi)-\overline{h}(\varphi)$, with
$h(\varphi,\psi)$ written according to (7.41), and estimate the whole average.
Recall that we are working with the extended maps (see Remark 7.20). After
rewriting $\langle h(\varphi,\cdot)-\overline{h}(\varphi)\rangle$ as

$\big{\langle}{h_{1}(\varphi,\cdot)-\overline{h}(\varphi)}\big{\rangle}+\big{\langle}{W(\cdot)\,\partial_{\varphi}h_{1}(\varphi,\cdot)}\big{\rangle}+\\!\int_{0}^{1}\\!\\!{\rm
d}t\,(1-t)\,\big{\langle}{(W(\cdot))^{2}\,\partial_{\varphi}^{2}h_{1}(\varphi+tW(\cdot),\cdot)}\big{\rangle},$

we bound $|\langle h_{1}(\varphi,\cdot)-\overline{h}(\varphi)\rangle|\leq
C\rho$ by Proposition 7.21, and $\langle(W(\cdot))^{2}\rangle\leq C\rho$ by
Theorem 4. Writing

$\big{\langle}{W(\cdot)\,\partial_{\varphi}h_{1}(\varphi,\cdot)}\big{\rangle}=\left\langle
W(\cdot)\,\partial_{\varphi}(h_{1}(\varphi,\cdot)-\overline{h}(\varphi))\right\rangle+\langle
W\rangle\partial_{\varphi}\overline{h}(\varphi),$

Theorem 4 gives $|\langle W\rangle\partial_{\varphi}\overline{h}(\varphi)|\leq
C\rho$ while we use first the Cauchy-Schwarz inequality, then Proposition
7.21, together again with Theorem 4, to obtain

$\left|\big{\langle}{W(\cdot)\,\partial_{\varphi}\left(h_{1}(\varphi,\cdot)-\overline{h}(\varphi)\right)}\big{\rangle}\right|\leq\left(\big{\langle}{(W(\cdot))^{2}}\big{\rangle}\big{\langle}\left(\partial_{\varphi}h_{1}(\varphi,\cdot)-\partial_{\varphi}\overline{h}(\varphi)\right)^{2}\big{\rangle}\right)^{\frac{1}{2}}\leq
C\rho.$ (7.105)

This concludes the proof of the first bound in (2.42a).

The second bound in (2.42a) is easily obtained by writing

$\displaystyle h(\varphi,\psi)-\overline{h}(\varphi)$
$\displaystyle=h(\varphi-W(\psi),\psi)-\overline{h}(\varphi)-W(\psi)\int_{0}^{1}{\rm
d}t\,\partial_{\varphi}h(\varphi-tW(\psi),\psi)$
$\displaystyle=h_{1}(\varphi,\psi)-\overline{h}(\varphi)-W(\psi)\int_{0}^{1}{\rm
d}t\,\partial_{\varphi}h(\varphi-tW(\psi),\psi),$

which implies

$\displaystyle\left(h(\varphi,\psi)-\overline{h}(\varphi)\right)^{2}$
$\displaystyle\leq
2\left(\left(h_{1}(\varphi,\psi)-\overline{h}(\varphi)\right)^{2}+(W(\psi))^{2}\left\|\partial_{\varphi}h\right\|^{2}\right)$
$\displaystyle\leq
4\left(h_{1}(\varphi,\psi)-h_{2}(\varphi,\psi)\right)^{2}+4\left(h_{2}(\varphi,\psi)-\overline{h}(\varphi)\right)^{2}+2(W(\psi))^{2}\left\|\partial_{\varphi}h\right\|^{2}$

which in turn is bounded using once more Proposition 7.21 and Theorem 4.

####  7.8.6 Deviations of the conjugation II: proof of the bounds (2.42b)

In order to study the average of
$\partial_{\varphi}h(\varphi,\psi)-\partial_{\varphi}\overline{h}(\varphi)$,
we write

$\displaystyle\partial_{\varphi}h(\varphi,\psi)$
$\displaystyle=\partial_{\varphi}h_{1}(\varphi,\psi)-W(\psi)\int_{0}^{1}{\rm
d}t\,\partial_{\varphi}^{2}h(\varphi-tW(\psi),\psi)$ (7.106)
$\displaystyle=\partial_{\varphi}h_{1}(\varphi,\psi)-W(\psi)\left(\partial_{\varphi}^{2}h_{1}(\varphi,\psi)-\partial_{\varphi}^{2}\overline{h}(\varphi)\right)$
$\displaystyle\qquad-W(\psi)\,\partial_{\varphi}^{2}\overline{h}(\varphi)+(W(\psi))^{2}\int_{0}^{1}{\rm
d}t\,\partial_{\varphi}^{3}h(\varphi-tW(\psi),\psi).$

Therefore, using the second expansion in (7.106), we can bound

$\displaystyle\left|\left\langle\partial_{\varphi}h(\varphi,\cdot)-\partial_{\varphi}\overline{h}(\varphi)\right\rangle\right|\leq\left|\left\langle\partial_{\varphi}h_{1}(\varphi,\cdot)-\partial_{\varphi}\overline{h}(\varphi)\right\rangle\right|$
(7.107) $\displaystyle\qquad+\left|\left\langle
W(\cdot)\left(\partial_{\varphi}^{2}h_{1}(\varphi,\cdot)-\partial_{\varphi}^{2}\overline{h}(\varphi)\right)\right\rangle\right|+\left\langle
W(\psi)\right\rangle\partial_{\varphi}^{2}\overline{h}(\varphi)+\left\langle(W(\psi))^{2}\right\rangle\|\partial_{\varphi}^{3}h\|_{\infty}.$

The term in the first line and the last two terms in the second line are
bounded by (2.42a) and by Theorem 4, respectively. To bound the first term in
the second line of (7.107) we need the following result, which is proved in
Appendix D.4.

######  Lemma 7.52.

Let $h_{1}$ and $\overline{h}$ be defined as in (6.18) and in (7.34),
respectively. Then, for all $\theta\in{\mathcal{U}}$, one has

$|\left\langle(\partial_{\theta}^{2}h_{1}(\theta,\cdot)-\partial_{\theta}^{2}\overline{h}(\theta))^{2}\right\rangle|\leq
C\rho.$

######  Remark 7.53.

_As mentioned in Remark 7.51, the condition $F\in{\mathscr{B}}_{\alpha_{0},6}$
is required to obtain the bounds in Lemma 7.52. A bound like
$|\left\langle\partial_{\theta}^{2}h_{1}(\theta,\cdot)-\partial_{\theta}^{2}\overline{h}(\theta)\right\rangle|\leq
C\rho$ could be obtained as well, but actually we need only the bound on the
squared deviations because the latter will be needed in order to estimate the
deviations of first derivative of the inverse conjugation. _

By using the Cauchy-Schwarz inequality and applying Lemma 7.52, we get

$\left|\left\langle
W(\cdot)\left(\partial_{\varphi}^{2}h_{1}(\varphi,\cdot)-\partial_{\varphi}^{2}\overline{h}(\varphi)\right)\right\rangle\right|\leq
C\rho,$

which inserted into (7.107) yields the first bound in (2.42b).

The second bound in (2.42b) is obtained by using the first expansion in
(7.106), which allows us to write

$\left|\partial_{\varphi}h(\varphi,\psi)-\partial_{\varphi}\overline{h}(\varphi)\right|\leq\left|\partial_{\varphi}h_{1}(\varphi,\psi)-\partial_{\varphi}\overline{h}(\varphi)\right|+|W(\psi)|\,\|\partial_{\varphi}^{2}h\|_{\infty}$

and hence

$\left\langle\left(\partial_{\varphi}h(\varphi,\cdot)-\partial_{\varphi}\overline{h}(\varphi)\right)^{2}\right\rangle\leq
2\left\langle\left(\partial_{\varphi}h_{1}(\varphi,\cdot)-\partial_{\varphi}\overline{h}(\varphi)\right)^{2}\right\rangle+2\|\partial_{\varphi}^{2}h\|_{\infty}^{2}\langle(W(\cdot))^{2}\rangle,$

which immediately implies the second bound in (2.42b) by (7.63) and by Theorem
4.

###  7.9 Deviations of the inverse conjugation

To deal with the inverse conjugation, we exploit the following trivial
identity.

${\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)=\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))=\eta,$
(7.108)

which holds for all
$(\eta,\psi)\in\Omega_{0}={\mathcal{U}}_{0}\times\mathds{T}^{2}:=\overline{\mathscr{H}}(\Omega)$,
provided $\Omega_{\rm ext}$ is such that ${\mathscr{H}}(\Omega_{\rm
ext})\supset\overline{\mathscr{H}}(\Omega)$ and $\Omega_{1,\rm
ext}\supset\Omega$ (recall that we are working with the extended maps).

######  Lemma 7.54.

Let $l_{1}$ and $\overline{l}$ be defined as in (6.19) and in (2.35),
respectively. Then, for all $\eta\in{\mathcal{U}}_{0}$, one has

$|\left\langle l_{1}(\eta,\cdot)-\overline{l}(\eta)\right\rangle|\leq
C\rho,\qquad\left\langle(l_{1}(\eta,\cdot)-\overline{l}(\eta))^{2}\right\rangle\leq
C\rho\,.$ (7.109)

###### Proof.

By using (7.108) we get

${\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)-{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta),\psi)=\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))-{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta),\psi).$
(7.110)

If we write in (7.110)

$\displaystyle{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)-{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta),\psi)$
$\displaystyle\qquad=\bigl{(}{\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}\int_{0}^{1}{\rm
d}t\,\partial_{\theta}{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta)+t({\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)),\psi)$

and use the lower bound (6.42), we obtain

$\gamma_{1}^{2}\big{\langle}{\bigl{(}{\mathcal{H}}_{1}^{-1}-\overline{\mathcal{H}}^{-1}\bigr{)}^{2}}\big{\rangle}(\eta)\leq\big{\langle}{\left({\mathcal{H}}_{1}-\overline{\mathcal{H}}\right)^{2}}\big{\rangle}\bigl{(}\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}.$
(7.111)

Analogously, if we write in (7.110)

$\displaystyle{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)-{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta),\psi)=\bigl{(}{\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\phantom{\int_{0}^{1}}$
$\displaystyle\qquad+\bigl{(}{\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}\left(\partial_{\theta}{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta),\psi)-\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\right)$
$\displaystyle\qquad+\bigl{(}{\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}^{2}\int_{0}^{1}(1-t)\,{\rm
d}t\,\partial_{\theta}^{2}{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta)+t({\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)),\psi),$

then we find

$\displaystyle\overline{\gamma}\bigl{|}\big{\langle}{{\mathcal{H}}_{1}^{-1}-\overline{\mathcal{H}}^{-1}}\big{\rangle}(\eta)\bigr{|}$
$\displaystyle\leq\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\bigl{|}\big{\langle}{{\mathcal{H}}_{1}^{-1}-\overline{\mathcal{H}}^{-1}}\big{\rangle}(\eta)\bigr{|}$
(7.112)
$\displaystyle\leq\bigl{|}\big{\langle}{{\mathcal{H}}_{1}-\overline{\mathcal{H}}}\big{\rangle}\bigl{(}\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}\bigr{|}$
$\displaystyle+\|\partial^{2}_{\theta}{\mathcal{H}}_{1}\|_{\infty}\big{\langle}{\bigl{(}{\mathcal{H}}_{1}^{-1}-\overline{\mathcal{H}}^{-1}\bigr{)}^{2}}\big{\rangle}(\eta)$
$\displaystyle+\Bigl{(}\big{\langle}{\bigl{(}{\mathcal{H}}_{1}^{-1}-\overline{\mathcal{H}}^{-1}\bigr{)}^{2}}\big{\rangle}(\eta)\big{\langle}{\bigl{(}\partial_{\theta}{\mathcal{H}}_{1}-\partial_{\theta}\overline{\mathcal{H}}\bigr{)}^{2}}\big{\rangle}\bigl{(}\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}\Bigr{)}^{1/2}\,,$

where $\overline{\gamma}=O(1)$ is a lower bound on
$\partial_{\theta}\overline{\mathcal{H}}$ (see Remark 6.15).

Thus, using that

${\mathcal{H}}_{1}(\theta,\psi)-\overline{\mathcal{H}}(\theta)=\theta^{2}\bigl{(}h_{1}(\theta,\psi)-\overline{h}(\theta)\bigr{)},\qquad{\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)=\eta^{2}\bigl{(}l_{1}(\eta,\psi)-\overline{l}(\eta)\bigr{)}$

and that there exists a constant $d_{0}$ such that
$d_{0}^{-1}\leq\theta/\eta\leq d_{0}$ for
$\theta=\overline{\mathcal{H}}^{-1}(\eta)$, as discussed in Section 6.4, the
bound (7.111), together with the second bound in (7.42a), gives the second
bound in (7.109), which, in turn, inserted into (7.112), together with the
first bound in (7.42a) and the second bound in (7.42b), yields the first bound
in (7.109). ∎

######  Remark 7.55.

_As the proof of Lemma 7.54 shows, in order to control the deviations of
$l_{1}-\overline{l}$, we need an estimate on the squared deviations of
$\partial_{\theta}h_{1}-\partial_{\theta}\overline{h}$, so that, even if we
confined ourselves to the deviations of the conjugations and the inverse
conjugations, in order to deal with the inverse conjugations we would need to
study the derivatives of $h$, and hence we should require $F$ to be in
${\mathscr{B}}_{\alpha_{0},5}$ and not only in ${\mathscr{B}}_{\alpha_{0},4}$.
In the same way, in order to study the deviations of the derivatives of the
inverse conjugation, we will need to control the squared deviations of
$\partial_{\theta}^{2}h_{1}-\partial_{\theta}^{2}\overline{h}$ through Lemma
7.52 and hence to require $F$ to be in ${\mathscr{B}}_{\alpha_{0},6}$ (see
Remark 7.53). _

######  Lemma 7.56.

Let $l_{1}$ and $\overline{l}$ be defined as in (6.19) and in (2.35),
respectively. Then, for all $\eta\in{\mathcal{U}}_{0}$, one has

$|\left\langle\partial_{\eta}l_{1}(\eta,\cdot)-\partial_{\eta}\overline{l}(\eta)\right\rangle|\leq
C\rho,\qquad\left\langle(\partial_{\eta}l_{1}(\eta,\cdot)-\partial_{\eta}\overline{l}(\eta))^{2}\right\rangle\leq
C\rho\,.$ (7.113)

###### Proof.

Differentiating (7.108) with respect to $\eta$, we obtain

$\partial_{\theta}{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)\,\partial_{\eta}{\mathcal{H}}_{1}^{-1}(\eta,\psi)=\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\,\partial_{\eta}\overline{\mathcal{H}}^{-1}(\eta)=1,\vspace{-.2cm}$
(7.114)

that implies

$\partial_{\eta}{\mathcal{H}}_{1}^{-1}(\eta,\psi)-\partial_{\eta}\overline{\mathcal{H}}^{-1}(\eta)=-\frac{\partial_{\theta}{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)-\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))}{\partial_{\theta}{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)\,\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))}.\vspace{-.2cm}$
(7.115)

Taking the square, we find

$\displaystyle\bigl{(}\partial_{\eta}{\mathcal{H}}_{1}^{-1}(\eta,\psi)-\partial_{\eta}\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}^{2}$
$\displaystyle\qquad\leq\frac{2}{\tau_{1}^{2}\overline{\tau}^{2}}\left(\|\partial_{\theta}^{2}{\mathcal{H}}_{1}\|_{\infty}^{2}\bigl{(}{\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}^{2}+\bigl{(}\partial_{\theta}{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta),\psi)-\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\bigr{)}^{2}\right),$

where $\gamma_{1}$ and $\overline{\gamma}$ are on lower bounds, respectively,
on $\partial_{\theta}{\mathcal{H}}_{1}$ and
$\partial_{\theta}\overline{\mathcal{H}}$ (see Subsection 6.4), so that

$\big{\langle}{\bigl{(}\partial_{\eta}l_{1}-\partial_{\eta}\overline{l}\bigr{)}^{2}}\big{\rangle}\leq\frac{2}{\tau_{1}^{2}\overline{\tau}^{2}}\left(\|\partial_{\theta}^{2}{\mathcal{H}}_{1}\|_{\infty}^{2}\big{\langle}{\bigl{(}l_{1}-\overline{l}\bigr{)}^{2}}\big{\rangle}+\big{\langle}{\bigl{(}\partial_{\theta}h_{1}-\partial_{\theta}\overline{h}\bigr{)}^{2}}\big{\rangle}\right),$

which, together with the second bound in (7.42b) and the second bound in
(7.109), implies the second of (7.113). Rewriting (7.115) as

$\displaystyle\partial_{\eta}{\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)-\partial_{\eta}\overline{\mathcal{H}}^{-1}(\eta)$
$\displaystyle\qquad=-\frac{\partial_{\theta}{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)-\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))}{\bigl{(}\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\bigr{)}^{2}}+\frac{\bigl{(}\partial_{\theta}{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)-\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\bigr{)}^{2}}{\partial_{\theta}{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)\,\bigl{(}\partial_{\theta}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\bigr{)}^{2}},$

and expanding

$\displaystyle\partial_{\theta}{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)=\partial_{\theta}{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta),\psi)$
$\displaystyle\qquad+\partial_{\theta}^{2}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\,\left(({\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\right)$
$\displaystyle\qquad+\left(\partial_{\theta}^{2}{\mathcal{H}}_{1}(\overline{\mathcal{H}}^{-1}(\eta),\psi)-\partial_{\theta}^{2}\overline{\mathcal{H}}(\overline{\mathcal{H}}^{-1}(\eta))\right)\,\left(({\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\right)$
$\displaystyle\qquad+\bigl{(}({\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}^{2}\int_{0}^{1}{\rm
d}t\,(1-t)\partial_{\theta}^{3}{\mathcal{H}}_{1}\bigl{(}\overline{\mathcal{H}}^{-1}(\eta),\psi)+t\bigl{(}({\mathcal{H}}_{1}^{-1}(\eta,\psi)-\overline{\mathcal{H}}^{-1}(\eta)\bigr{)}\bigr{)},$

we obtain eventually

$\displaystyle\bigl{|}\big{\langle}{\partial_{\eta}l_{1}-\partial_{\eta}\overline{l}(\eta)}\big{\rangle}\bigr{|}$
$\displaystyle\leq\frac{1}{\overline{\tau}^{2}}\Bigl{(}\bigl{|}\big{\langle}{\partial_{\theta}h_{1}-\partial_{\theta}\overline{h}}\big{\rangle}\bigr{|}+\|\partial_{\theta}^{2}\overline{\mathcal{H}}\|_{\infty}\bigl{|}\langle
l_{1}-\overline{l}\rangle\bigr{|}+\bigl{(}\big{\langle}{(\partial_{\theta}^{2}h_{1}-\partial_{\theta}^{2}\overline{h})^{2}}\big{\rangle}\langle(l_{1}-\overline{l})^{2}\rangle\bigr{)}^{1/2}$
$\displaystyle+\|\partial_{\theta}^{2}{\mathcal{H}}_{1}\|_{\infty}\bigl{|}\langle(l_{1}-\overline{l})^{2}\rangle\bigr{|}\Bigr{)}+\frac{2}{\tau_{1}^{2}\overline{\tau}}\Bigl{(}\|\partial_{\theta}^{3}{\mathcal{H}}_{1}\|_{\infty}\bigl{|}\langle(l_{1}-\overline{l})^{2}\rangle\bigr{|}+\bigl{|}\big{\langle}{(\partial_{\theta}h_{1}-\partial_{\theta}\overline{h})^{2}}\big{\rangle}\bigr{|}\Bigr{)},$

so that the first of (7.113) follows by using the previous estimates of
Propositions 7.21 and Lemmas 7.52 and 7.56. ∎

Recalling that $l=l_{1}$, by (6.20), Lemmas 7.54 and 7.56 immediately imply
the following result, which is the analogue of Theorem 5 for the inverse
conjugation.

######  Proposition 7.57.

Let $l$ be defined as in (2.25). Then there is a constant $C$ such that, for
all $\eta\in{\mathcal{U}}_{0}$,

$\displaystyle\left|\left\langle
l(\eta,\cdot)-\overline{l}(\eta)\right\rangle\right|$ $\displaystyle\leq
C\rho,$
$\displaystyle\left\langle(l(\eta,\cdot)-\overline{h}(\eta))^{2}\right\rangle$
$\displaystyle\leq C\rho$ (7.116a)
$\displaystyle\left|\left\langle\partial_{\eta}l(\eta,\cdot)-\partial_{\eta}\overline{l}(\eta)\right\rangle\right|$
$\displaystyle\leq C\rho,$
$\displaystyle\left\langle(\partial_{\eta}l(\eta,\cdot)-\partial_{\eta}\overline{l}(\eta))^{2}\right\rangle$
$\displaystyle\leq C\rho\,.$ (7.116b)

###  7.10 Fluctuations of the dynamics: proof of Theorem 6

We start by proving the following result for the extension of the translated
map ${\mathscr{S}}_{1}$.

######  Lemma 7.58.

For any $\gamma^{\prime}\in(0,\gamma)$, there exists a constant $C$ such that,
for all $\theta\in{\mathcal{U}}$ and all $n\in\mathds{N}$, one has

$\bigl{|}\big{\langle}{({\mathscr{S}}_{1}^{n})_{\theta}(\theta,\cdot)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta)}\big{\rangle}\bigr{|}\leq
C\rho(1-\rho\,\gamma^{\prime})^{n},\qquad\bigl{\langle}\bigl{(}({\mathscr{S}}_{1}^{n})_{\theta}(\theta,\cdot)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta)\bigr{)}^{2}\bigr{\rangle}\leq
C\rho\,(1-\rho\,\gamma^{\prime})^{2n}.$

Proof. From (2.34), (6.17) and (6.19), we find

$\displaystyle({\mathscr{S}}_{1}^{n})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta)={\mathcal{H}}_{1}^{-1}({\mathscr{S}}_{0}^{n}({\mathcal{H}}_{1}(\theta,\psi),\psi))-\overline{\mathcal{H}}^{-1}(\overline{\mathscr{S}}_{0}^{n}(\overline{\mathcal{H}}(\theta),\psi))$
(7.117)
$\displaystyle\qquad\qquad=\kappa^{(n)}(\psi)\,{\mathcal{H}}_{1}(\theta,\psi)-\overline{\mu}^{n}\overline{\mathcal{H}}(\theta)$
$\displaystyle\qquad\qquad+\Bigl{(}\bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}_{1}(\theta,\psi)\bigr{)}^{2}-\bigl{(}\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\bigr{)}^{2}\Bigr{)}\,\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)}$
$\displaystyle\qquad\qquad+\Bigl{(}\bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}_{1}(\theta,\psi)\bigr{)}^{2}-\bigl{(}\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\bigr{)}^{2}\Bigr{)}\Bigl{(}l_{1}\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi),A_{0}^{n}\psi\bigr{)}-\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)}$
$\displaystyle\qquad\qquad+\bigl{(}\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\bigr{)}^{2}\Bigl{(}l_{1}\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi),A_{0}^{n}\psi\bigr{)}-\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)},$

where, after further expanding, in the third and fourth line,

$\displaystyle\bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}_{1}(\theta,\psi)\bigr{)}^{2}-\bigl{(}\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\bigr{)}^{2}$
(7.118)
$\displaystyle\qquad\qquad=\Bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}_{1}(\theta,\psi)-\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\Bigr{)}^{2}+2\,\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\,\Bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}_{1}(\theta,\psi)\bigr{)}-\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\Bigr{)},$

and, in the fourth and fifth line,

$\displaystyle
l_{1}\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi),A_{0}^{n}\psi\bigr{)}-\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}$
(7.119)
$\displaystyle\quad=\partial_{\eta}\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\,\Bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi)-\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\Bigr{)}$
$\displaystyle\quad+\bigl{(}\partial_{\eta}l_{1}\bigl{(}\overline{\mu}^{n}\overline{\mathcal{H}}(\theta),A_{0}^{n}\psi\bigr{)}-\partial_{\eta}\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\,\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi)-\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}$
$\displaystyle\quad+\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi)\\!-\\!\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}^{2}$
$\displaystyle\qquad\qquad\times\\!\\!\\!\int_{0}^{1}\\!\\!\\!{\rm
d}t\,(1\\!-\\!t)\,\partial_{\eta}^{2}l_{1}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)+t\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi)\\!-\\!\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)},A_{0}^{n}\psi\bigr{)}$
$\displaystyle\quad+l_{1}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta),A_{0}^{n}\psi\bigr{)}-\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)},$

we write

$\displaystyle\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi)-\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)$
(7.120)
$\displaystyle\quad=\bigl{(}\kappa^{(n)}(\psi)-\overline{\mu}^{n}\bigr{)}\,\bigl{(}{\mathcal{H}}_{1}(\theta,\psi)-\overline{\mathcal{H}}(\theta)\bigr{)}+\bigl{(}\kappa^{(n)}(\psi)-\overline{\mu}^{n}\bigr{)}\,\overline{\mathcal{H}}(\theta)+\overline{\mu}^{n}\bigl{(}{\mathcal{H}}_{1}(\theta,\psi)-\overline{\mathcal{H}}(\theta)\bigr{)}$
$\displaystyle\quad=\theta^{2}\bigl{(}\kappa^{(n)}(\psi)-\overline{\mu}^{n}\bigr{)}\,\bigl{(}h_{1}(\theta,\psi)-\overline{h}(\theta)\bigr{)}+\bigl{(}\kappa^{(n)}(\psi)-\overline{\mu}^{n}\bigr{)}\,\overline{\mathcal{H}}(\theta)+\theta^{2}\overline{\mu}^{n}\bigl{(}h_{1}(\theta,\psi)-\overline{h}(\theta)\bigr{)}.$

Thus, using Lemma 2.33, Propositions 7.21 and 7.56, and the Cauchy-Schwarz
inequality, the first bound follows.

The second bound is obtained in a similar way. Shortening, for notational
simplicity,

$\displaystyle{\mathcal{A}}_{n}(\theta,\psi)$
$\displaystyle:=\kappa^{(n)}{\mathcal{H}}_{1}(\theta,\psi)-\overline{\mu}^{n}\;\overline{\mathcal{H}},$
$\displaystyle\lambda_{1}(\theta,\psi)$
$\displaystyle:=l_{1}\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi),A_{0}^{n}\psi\bigr{)},\qquad\lambda_{2}(\theta,\psi):=\partial_{\eta}l_{1}\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi),A_{0}^{n}\psi\bigr{)},$
$\displaystyle\bar{\lambda}(\theta)$
$\displaystyle:=\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)},\hskip
81.65926pt\bar{\lambda}_{1}(\theta,\psi):=l_{1}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta),A_{0}^{n}\psi\bigr{)}\Bigr{)},$
$\displaystyle I(\theta,\psi)$
$\displaystyle:=\\!\\!\\!\int_{0}^{1}\\!\\!\\!{\rm
d}t\,(1\\!-\\!t)\,\partial_{\eta}^{2}l_{1}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)+t\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}_{1}(\theta,\psi)\\!-\\!\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)},A_{0}^{n}\psi\bigr{)}$

and using (7.118), we can rewrite (7.117) as

$({\mathscr{S}}_{1}^{n})_{\theta}-(\overline{\mathscr{S}}^{n})_{\theta}={\mathcal{A}}_{n}+\bigl{(}{\mathcal{A}}_{n}+2\,\overline{\mu}^{n}\,\overline{\mathcal{H}}\,{\mathcal{A}}_{n}\bigr{)}^{2}\,\lambda_{1}+\bigl{(}\overline{\mu}^{n}\,\overline{\mathcal{H}}\bigr{)}^{2}\bigl{(}\lambda_{1}-\bar{\lambda}\bigr{)},$

and (7.119) as

$\lambda_{1}-\bar{\lambda}=\lambda_{2}{\mathcal{A}}_{n}+I_{n}\,{\mathcal{A}}_{n}^{2}+\bigl{(}\bar{\lambda}_{1}-\bar{\lambda}\bigr{)},$

so that, bounding

$\displaystyle\bigl{(}({\mathscr{S}}_{1}^{n})_{\theta}-(\overline{\mathscr{S}}^{n})_{\theta}\bigr{)}^{2}$
$\displaystyle\leq
3\left({\mathcal{A}}_{n}^{2}+2\,{\mathcal{A}}_{n}^{2}\bigl{(}{\mathcal{A}}_{n}^{2}+4\,(\overline{\mu}^{n}\,\overline{\mathcal{H}})^{2}\bigr{)}\lambda_{1}^{2}+\bigl{(}\overline{\mu}^{n}\,\overline{\mathcal{H}}\bigr{)}^{4}\bigl{(}\lambda_{1}-\bar{\lambda}\bigr{)}^{2}\right),$
$\displaystyle\bigl{(}\lambda_{1}-\bar{\lambda}\bigr{)}^{2}$
$\displaystyle\leq
3\left(\lambda_{2}^{2}{\mathcal{A}}_{n}^{2}+I_{n}^{2}\,{\mathcal{A}}_{n}^{4}+\bigl{(}\bar{\lambda}_{1}-\bar{\lambda}\bigr{)}^{2}\right),$

and, thanks to (7.120),

${\mathcal{A}}_{n}^{2}\leq
2\left(\bigl{(}\kappa^{(n)}-\overline{\mu}^{n}\bigr{)}^{2}\overline{\mathcal{H}}+\theta^{4}\overline{\mu}^{2n}\bigl{(}h_{1}-\overline{h}\bigr{)}^{2}\right),$

we obtain

$\displaystyle\big{\langle}{\bigl{(}({\mathscr{S}}_{1}^{n})_{\theta}(\theta,\cdot)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta)\bigr{)}^{2}}\big{\rangle}$
$\displaystyle\qquad\leq
C\Bigl{(}\bigl{(}\big{\langle}{\kappa^{(n)}(\cdot)-\overline{\mu}^{n}\bigr{)}^{2}}\big{\rangle}+\overline{\mu}^{2n}\big{\langle}{\bigl{(}h_{1}(\theta,\cdot)-\bar{h}(\theta)\bigr{)}^{2}}\big{\rangle}+\overline{\mu}^{4n}\big{\langle}{\bigl{(}l_{1}(\overline{\mu}^{n}\overline{\mathcal{H}}(\theta),\cdot)-\bar{l}(\overline{\mu}^{n}\overline{\mathcal{H}}(\theta)\bigr{)}^{2}}\big{\rangle}\Bigr{)}.$

Then, using Lemma 2.33 and Propositions 7.21 and 7.56 gives immediately the
second bound. ∎

Finally, we come back to the map ${\mathscr{S}}$. In order to compare the full
dynamics generated by ${\mathscr{S}}$ with that generated by
$\overline{\mathscr{S}}$, we proceed along the same lines as the proof of
Lemma 7.58. Thus, we decompose
$({\mathscr{S}}^{n}\big{)}_{\varphi}(\varphi,\psi)-(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi)$
as done for
$({\mathscr{S}}_{1}^{n}\big{)}_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta)$,
relying on (2.23) and (2.25) instead of (6.17) and (6.19), and obtain

$\displaystyle({\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)-(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi)={\mathcal{H}}^{-1}({\mathscr{S}}_{0}^{n}({\mathcal{H}}(\theta,\psi),\psi))-\overline{\mathcal{H}}^{-1}(\overline{\mathscr{S}}_{0}^{n}(\overline{\mathcal{H}}(\theta),\psi))$
$\displaystyle\quad=W(A_{0}^{n}\psi)+\kappa^{(n)}(\psi)\,{\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)+\Bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\Bigr{)}^{2}\,\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)}$
$\displaystyle\quad+\Bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\Bigr{)}^{2}\,\Bigl{(}l\bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}(\theta,\psi),\psi\bigr{)}-\overline{l}\big{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)}$
$\displaystyle\quad+2\,\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\,\Bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\Bigr{)}\,\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)}$
$\displaystyle\quad+2\,\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\,\Bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\Bigr{)}\,\Bigl{(}l\bigl{(}\kappa^{(n)}(\psi)\,{\mathcal{H}}(\theta,\psi),\psi\bigr{)}-\overline{l}\big{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)}$
$\displaystyle\quad+\bigl{(}\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)\bigr{)}^{2}\Bigl{(}\partial_{\eta}\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\,\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}$
$\displaystyle\quad+\bigl{(}\partial_{\eta}l\bigl{(}\overline{\mu}^{n}\overline{\mathcal{H}}(\theta),A_{0}^{n}\psi\bigr{)}\bigr{)}-\partial_{\eta}\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\,\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}$
$\displaystyle\quad+\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}^{2}\\!\\!\int_{0}^{1}\\!\\!\\!{\rm
d}t\,(1\\!-\\!t)\,\partial_{\eta}^{2}l\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)+t\bigl{(}\kappa^{(n)}(\psi){\mathcal{H}}(\theta,\psi)\\!-\\!\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)},A_{0}^{n}\psi\bigr{)}$
$\displaystyle\quad+l\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta),A_{0}^{n}\psi\bigr{)}-\overline{l}\bigl{(}\overline{\mu}^{n}\;\overline{\mathcal{H}}(\theta)\bigr{)}\Bigr{)},$

where we expand

$\displaystyle\kappa^{(n)}(\psi)\,{\mathcal{H}}(\theta,\psi)-\overline{\mu}^{n}\,\overline{\mathcal{H}}(\theta)$
$\displaystyle\quad=\theta^{2}\bigl{(}\kappa^{(n)}(\psi)-\overline{\mu}^{n}\bigr{)}\,\bigl{(}h(\theta,\psi)-\overline{h}(\theta)\bigr{)}+\bigl{(}\kappa^{(n)}(\psi)-\overline{\mu}^{n}\bigr{)}\,\overline{\mathcal{H}}(\theta)+\theta^{2}\overline{\mu}^{n}\bigl{(}h(\theta,\psi),A_{0}^{n}\psi\bigr{)}-\overline{h}(\theta)\bigr{)}.$

Then, using the bounds of Lemma 2.33, Theorem 4, Proposition 7.57, and the
Cauchy-Schwarz inequality, we obtain the bound (2.44a) of Theorem 6. Again,
the bound (2.44b) is obtained by reasoning in a similar way.

##  8 Convergence in probability

In this section we collect the previous results to prove Theorems 7 and 8, so
as to provide the probabilistic description of the results discussed in
Subsection 2.4.5. Below, as in Subsections 7.8 to 7.10, for notational
simplicity, ${\mathscr{S}}_{1}$ denotes the extended map
${\mathscr{S}}_{1,{\rm ext}}$.

###  8.1 The probability of deviations: proof of Theorem 7

From Theorem 1 and a direct application of Chebyshev inequality we obtain
that, for any $\delta>0$,

$m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:|W(\psi)|>\delta\Bigr{\\}}\Bigr{)}\leq\frac{C\rho}{\delta^{2}},$
(8.1)

for a suitable positive constant $C$ independent of $n$. Therefore, the
invariant manifold ${\mathcal{W}}$ for ${\mathscr{S}}$ converges in
probability to $\overline{\mathcal{W}}=\\{(0,\psi):\psi\in\mathds{T}^{2}\\}$.

Similarly, from Lemma 7.58, we obtain that, if $\gamma^{\prime}\in(0,\gamma)$,
then, for fixed $\theta$ and $n$, and for any $\delta>0$,

$m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}_{1}^{n})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|>\delta\Bigr{\\}}\Bigr{)}\leq\frac{C(\gamma^{\prime})\,\rho}{\delta^{2}},$
(8.2)

for a suitable positive constant $C(\gamma^{\prime})$ depending on
$\gamma^{\prime}$ but neither on $\theta$ nor $n$.

Note that the set of angles $\psi$ considered in (8.2) depends on $n$, even
though its measure is bounded independently of the value of $n$. The following
result provides a uniform version of the estimate above, and shows that, for
most values of $\psi\in\mathds{T}^{2}$, the quantity
$(1-\rho\,\gamma^{\prime})^{-n}|\,({\mathscr{S}}_{1}^{n})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|$
remains small for all $n\geq 0$.

######  Lemma 8.1.

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Let ${\mathscr{S}}_{1}$ be defined as in (6.13).
For $\rho$ small enough and any $\gamma^{\prime}\in(0,\gamma)$, there exists a
constant $C$ such that, for all $\theta\in{\mathcal{U}}$,

$m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:\sup_{n\geq
0}\,(1-\rho\,\gamma^{\prime})^{-n}|\,({\mathscr{S}}_{1}^{n})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|>\delta\Bigr{\\}}\Bigr{)}\leq\frac{C\rho}{\delta^{3}}.$

Proof. Using that
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|f|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},6}=1$,
for any $\theta,\theta^{\prime}\in{\mathcal{U}}$, we get

$|({\mathscr{S}}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}})_{\theta}(\theta^{\prime},\psi)|\geq(1-\rho)|\theta-\theta^{\prime}|-\rho|f_{1}(\theta,\psi)-\bar{f}(\theta)|\geq(1-\rho)|\theta-\theta^{\prime}|-C\rho|\theta|\,,$

where we have also used that $f_{1}(0,\psi)=\overline{f}(0)=0$, as it follows
from the definition of $f_{1}$ in (7.44) and from Remark 2.10. Iterating we
obtain, for any $\theta,\theta^{\prime}\in{\mathcal{U}}$ and for any $k\geq
0$,

$|({\mathscr{S}}_{1}^{k})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}_{\theta}^{k}(\theta^{\prime},\psi)|\geq(1-\rho)^{k}|\theta-\theta^{\prime}|-Ck(1-\rho\,\gamma^{\prime})^{k}\rho|\theta|,$

where we have used that $(1-\rho)<(1-\rho\,\gamma^{\prime})$, since
$\gamma^{\prime}<\gamma<1$ (see Remark 2.20), and that, by Remark 6.12,
$|({\mathscr{S}}_{1}^{k})_{\theta}(\theta,\psi)|\leq
C(1-\rho\,\gamma^{\prime})^{k}|\theta|$. Assuming that for some $n>0$ we have

$(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|>\delta,$

we find that

$(1-\rho\,\gamma^{\prime})^{-(n+k)}|({\mathscr{S}}^{n+k}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n+k})_{\theta}(\theta,\psi)|\geq\left(\frac{1-\rho}{1-\rho\,\gamma^{\prime}}\right)^{k}\delta-
Ck\rho\,.$

For $\rho$ small enough, we can now find $M\in\mathds{N}$ of the form
$M=M_{0}\delta/\rho$, with $M_{0}$ independent of $\rho$ and $\delta$, such
that, for all $k\leq M$,

$Ck\rho\leq\frac{\delta}{4},\qquad\hbox{and}\qquad\left(\frac{1-\rho}{1-\rho\,\gamma^{\prime}}\right)^{k}\geq\frac{1}{2}.$

This means that for $k\leq M$ we have

$(1-\rho\,\gamma^{\prime})^{-(n+k)}|({\mathscr{S}}^{n+k}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n+k})_{\theta}(\theta,\psi)|\geq\frac{\delta}{4}.$

We thus obtain that

$\displaystyle\sum_{n=0}^{\infty}m_{0}\left(\left\\{\psi\in\mathds{T}^{2}:(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|>\frac{\delta}{4}\right\\}\right)$
(8.3)
$\displaystyle\qquad\geq\frac{M_{0}\delta}{\rho}\,m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:\sup_{n\geq
0}\,(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|>\delta\Bigr{\\}}\Bigr{)}.$

On the other hand, one has

$\sum_{n=0}^{\infty}m_{0}\left(\left\\{\psi\in\mathds{T}^{2}:(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|>\frac{\delta}{4}\right\\}\right)\leq\frac{C_{0}(\gamma^{\prime})}{\delta^{2}},$
(8.4)

for a suitable constant $C_{0}(\gamma^{\prime})$ depending on
$\gamma^{\prime}$. This can be seen in the following way. For any
$\gamma^{\prime},\gamma^{\prime\prime}\in(0,\gamma)$ we find, by using (8.2),

$m_{0}\left(\left\\{\psi\in\mathds{T}^{2}:(1-\rho\,\gamma^{\prime\prime})^{-n}|({\mathscr{S}}^{n}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|>\frac{\delta}{4}\right\\}\right)\leq\left(\frac{1-\rho\,\gamma^{\prime\prime}}{1-\rho\,\gamma^{\prime}}\right)^{2n}\frac{16\,C(\gamma^{\prime})\,\rho}{\delta^{2}},$

with $C(\gamma^{\prime})$ as in (8.2). If we invert the roles of
$\gamma^{\prime}$ and $\gamma^{\prime\prime}$, we obtain

$m_{0}\left(\left\\{\psi\in\mathds{T}^{2}:(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)|>\frac{\delta}{4}\right\\}\right)\leq\left(\frac{1-\rho\,\gamma^{\prime}}{1-\rho\,\gamma^{\prime\prime}}\right)^{2n}\frac{16\,C(\gamma^{\prime\prime})\,\rho}{\delta^{2}},$

so that, fixing $\gamma^{\prime}\in(0,\gamma)$ and taking
$\gamma^{\prime\prime}=\gamma^{\prime}/2$, then (8.4) follows with

$C_{0}(\gamma^{\prime})=16\,C(\gamma^{\prime}/2)\,\rho\sum_{n=0}^{\infty}\left(\frac{1-\rho\,\gamma^{\prime}}{1-\rho\,\gamma^{\prime}/2}\right)^{2n}.$

Inserting (8.4) into (8.3) delivers the thesis. ∎

######  Remark 8.2.

_As a consequence of Lemma 8.1, for the overwhelming majority of initial
$\psi$, the dynamics generated by ${\mathscr{S}}_{1}$ is very well described
by the averaged dynamics generated by $\overline{\mathscr{S}}$ uniformly in
$n$. _

To deal with the map ${\mathscr{S}}$, we follow the same argument used for
${\mathscr{S}}_{1}$. First of all, from Theorem 6 and Chebyshev inequality, if
$\gamma^{\prime}\in(0,\gamma)$, then, for fixed $\theta$ and $n$, and for any
$\delta>0$,

$m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\varphi,\psi)-W(A_{0}^{n}\psi)|>\delta\Bigr{\\}}\Bigr{)}\leq\frac{C(\gamma^{\prime})\,\rho}{\delta^{2}},$

with the constant $C(\gamma^{\prime})$ independent of both $\theta$ and $n$.
Next, observe that we can write

$\displaystyle\bigl{|}{\mathscr{S}}_{\varphi}(\varphi,\psi)-\overline{\mathscr{S}}_{\varphi}(\varphi^{\prime},\psi)-W(A_{0}\psi)\bigr{|}=\bigl{|}{\mathscr{S}}_{\varphi}(\varphi,\psi)-\overline{\mathscr{S}}_{\varphi}(\varphi^{\prime},\psi)-W(\psi)-\rho\,f(W(\psi),\psi)\bigr{|}$
$\displaystyle\quad=\bigl{|}\varphi-W(\psi)-\varphi^{\prime}+\rho\bigl{(}f(\varphi,\psi)-\rho\,f(W(\psi),\psi)-\rho\,\overline{f}(\varphi-W(\psi)\bigr{)}+\rho\bigl{(}\overline{f}(\varphi-W(\psi))-\overline{f}(\varphi^{\prime})\bigr{)}\bigr{|}$
$\displaystyle\quad\geq(1-\rho)\bigl{|}\varphi-W(\psi)-\varphi^{\prime}\bigr{|}-C\rho\bigl{|}\varphi-W(\psi)\bigr{|},$

so that, iterating and using that

$\bigl{|}({\mathscr{S}}^{i})_{\varphi}(\varphi,\psi)-W(A_{0}^{i}\psi)\bigr{|}=\bigl{|}({\mathscr{S}}_{1}^{i})_{\theta}(\theta,\varphi)\bigl{|}\leq
C(1-\rho\,\gamma^{\prime})^{i}|\theta|,$

where $\theta=\varphi-W(\psi)$, we find, once again thanks to Lemma 6.11,

$\bigl{|}({\mathscr{S}}^{k})_{\varphi}(\varphi,\psi)-(\overline{\mathscr{S}}^{k})_{\varphi}(\varphi^{\prime},\psi)-W(A_{0}^{n}\psi)\bigr{|}\geq(1-\rho)^{k}\bigl{|}\varphi-W(\psi)-\varphi^{\prime}\bigr{|}-Ck(1-\rho\,\gamma^{\prime})^{k}\rho\bigl{|}\varphi-W(\psi)\bigr{|}.$

Therefore, assuming that for some $n>0$ we have

$(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)-W(A_{0}^{n}\psi)|>\delta$

and proceeding as done for ${\mathscr{S}}_{1}$ leads to

$\displaystyle\frac{C_{1}(\gamma^{\prime})}{\delta^{2}}$
$\displaystyle\geq\sum_{n=0}^{\infty}m_{0}\left(\left\\{\psi\in\mathds{T}^{2}:(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)-(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)-W(A_{0}^{n}\psi)|>\frac{\delta}{4}\right\\}\right)$
$\displaystyle\geq\frac{M_{0}\delta}{\rho}\,m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:\sup_{n\geq
0}\,(1-\rho\,\gamma^{\prime})^{-n}|({\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)-(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)-W(A_{0}^{n}\psi)|>\delta\Bigr{\\}}\Bigr{)}.$

for a suitable constant $C_{1}(\gamma^{\prime})$. This completes the proof of
Theorem 7.

###  8.2 Continuous time: proof of Theorem 8

From (6.16) we get

$\displaystyle\widetilde{X}_{t}(\psi)$
$\displaystyle=({\mathscr{S}}_{1}^{\lfloor
t/\rho\rfloor})_{\theta}(\varphi_{0}-W(\psi),\psi)$
$\displaystyle+(t/\rho-\lfloor t/\rho\rfloor)\left(({\mathscr{S}}_{1}^{\lfloor
t/\rho\rfloor+1})_{\theta}(\varphi_{0}-W(\psi),\psi)-({\mathscr{S}}_{1}^{\lfloor
t/\rho\rfloor})_{\theta}(\varphi_{0}-W(\psi),\psi)\right)\,.$

Since, by Lemma 6.11,

$\left|({\mathscr{S}}_{1}^{k})_{\theta}(\varphi_{0}-W(\psi),\psi)-({\mathscr{S}}_{1}^{k})_{\theta}(\varphi_{0},\psi)\right|\leq
C(1-\rho\,\gamma^{\prime})^{k}|W(\psi)|,$ (8.5)

by combining the bound (8.5) with Lemmas 7.58 and 2.29, we get, for
$\xi\in(0,\gamma)$,

$\sup_{t\geq 0}e^{\xi
t}|\widetilde{X}_{t}(\psi)-\Phi_{t}(\varphi_{0})|\leq\sup_{n\geq
0}\,(1-\rho\,\xi)^{-n}\left|({\mathscr{S}}^{n}_{1})_{\theta}(\theta,\psi)-(\overline{\mathscr{S}}^{n})_{\theta}(\theta,\psi)\right|+C\left(|W(\psi)|+\rho\right),$

so that, for every $\delta>0$, we have

$\lim_{\rho\to
0^{+}}\,m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:\sup_{t\geq 0}\,e^{\xi
t}|\widetilde{X}_{t}(\psi)-\Phi_{t}(\varphi_{0})|\geq\delta\Bigr{\\}}\Bigr{)}=0,$

because of (2.39) and Lemma 8.1.

##  Appendix A Decay of correlations for Hölder continuous functions

###  A.1 Symbolic dynamics

Consider a Markov partition ${\mathcal{Q}}=\\{Q_{1},Q_{2},\ldots,Q_{s}\\}$ for
the Anosov automorphism $A_{0}$ in (2.15), and call $T$ be the corresponding
compatibility matrix. For any Anosov automorphism $A_{0}$ on $\mathds{T}^{n}$,
the largest eigenvalue $\lambda_{1}$ of the compatibility matrix is simple (by
Perron-Frobenius Theorem) and equals the largest eigenvalue
$\lambda_{+}=\lambda$ of $A_{0}$ [29, Ch. 6]; a stronger result holds for
$n=2$ (see Remark A.2 below). Let $a$ denote the mixing time, i.e. the minimum
value $a\in\mathds{N}$ such that all the entries of $T^{a+1}$ are different
from $0$ (since every Anosov automorphism is transitive, $a$ is finite).

Set ${\mathcal{N}}=\\{1,2,\ldots,s\\}$. We say that a sequence
$\underline{\sigma}\in{\mathcal{N}}^{\mathds{Z}}$ is $T$-compatible if
$T_{\sigma_{k},\sigma_{k+1}}=1$ for all $k\in\mathds{Z}$; let
${\mathcal{N}}^{\mathds{Z}}_{T}$ denote the set of $T$-compatible sequences.
We use the symbol $\underline{\sigma}$ also to denote subsequences, i.e.
elements of ${\mathcal{N}}^{[n,m]}$, with finite $n$ and $m$. Given two
sequences $\underline{\sigma}$ and $\underline{\sigma}^{\prime}$, we set
$\nu(\underline{\sigma},\underline{\sigma}^{\prime}):=\max\\{k\in\mathds{N}:\sigma_{i}=\sigma^{\prime}_{i},|i|<k\\}$.
On ${\mathcal{N}}^{\mathds{Z}}_{T}$ we consider the topology generated by the
distance
$d(\underline{\sigma},\underline{\sigma}^{\prime})=\lambda^{-\nu(\underline{\sigma},\underline{\sigma}^{\prime})}$.
Given $\underline{\sigma}\in{\mathcal{N}}^{\mathds{Z}}_{T}$, let
$\psi=\psi(\underline{\sigma})\in\mathds{T}^{2}$ be the unique point whose
symbolic representation is $\underline{\sigma}$; then we have

$A_{0}\psi(\underline{\sigma})=\psi(\tau\underline{\sigma}),$

where $\tau$ is the left translation, that is
$(\tau\underline{\sigma})_{i}=\sigma_{i+1}$, and

$|\psi(\underline{\sigma})-\psi(\underline{\sigma}^{\prime})|\leq
C_{s}\,d(\underline{\sigma},\underline{\sigma}^{\prime}),$

where $C_{s}$ is the maximum diameter of the sets $Q_{1},\ldots,Q_{s}$.

We call $C_{\underline{\sigma}}^{J}$ the $T$-compatible cylinder with base
$J=(j_{1},\ldots,j_{q})\subset\mathds{Z}$ and specification
$\underline{\sigma}=(\sigma_{j_{1}},\ldots,\sigma_{j_{q}})\in{\mathcal{N}}^{q}_{T}$,
i.e. the set of sequences
$\underline{\sigma^{\prime}}\in{\mathcal{N}}^{\mathds{Z}}_{T}$ such that
$\sigma_{j_{k}}^{\prime}=\sigma_{j_{k}}$ for $k=1,\ldots,q$. In the following
we mainly consider sets $J$ such that $j_{1},\ldots,j_{q}$ are consecutive,
i.e. $j_{k+1}=j_{k}+1$ for $k=1,\ldots,q-1$; in that case we also write
$J=[j_{1},j_{q}]$ and $\sigma_{j_{k}}=\underline{\sigma}(j_{k})$ for
$k=1,\ldots,q$, and we say that the subsequence
$(\sigma_{j_{1}},\ldots,\sigma_{j_{q}})$ is $T$-compatible on $J$ if
$T_{\underline{\sigma}(j_{k})\underline{\sigma}(j_{k+1})}>0$ for
$k=1,\ldots,q-1$.

Finally call $m$ the Gibbs measure associated with the Lebesgue measure on
$\mathds{T}^{2}$ [29, Ch. 5]. If $\Sigma(M,\sigma,\sigma^{\prime})$ denotes
the set of $T$-compatible sequences $\underline{\sigma}$ on $[-M-1,M+1]$
starting with $\sigma$ and ending with $\sigma^{\prime}$, i.e. such that
$\underline{\sigma}(-M-1)=\sigma$ and
$\underline{\sigma}(M+1)=\sigma^{\prime}$, one has

$m(C_{\underline{\sigma}}^{J})=\lim_{M\to\infty}\frac{|\Sigma(M,\sigma,\sigma^{\prime})\cap
C_{\underline{\sigma}}^{J}|}{|\Sigma(M,\sigma,\sigma^{\prime})|},$ (A.1)

where the limit does not depend on $\sigma$ and $\sigma^{\prime}$ (which can
be fixed arbitrarily, say equal to 1).

######  Lemma A.1.

Let $n_{1},n_{2},n_{1}^{\prime},n_{2}^{\prime}\in\mathds{Z}$ such that
$n_{1}<n_{2}$, $n^{\prime}_{1}<n^{\prime}_{2}$ and $n^{\prime}_{1}-n_{2}>a$.
Consider two cylinders $C_{\underline{\sigma}_{1}}^{J_{1}}$ and
$C_{\underline{\sigma}_{2}}^{J_{2}}$, with $J_{1}=[n_{1},n_{2}]$ and
$J_{2}=[n^{\prime}_{1},n^{\prime}_{2}]$. Then one has

$1-K_{0}\lambda_{0}^{-(n^{\prime}_{1}-n_{2})}\leq\frac{m(C_{\underline{\sigma}_{1}}^{J_{1}}\cap
C_{\underline{\sigma}_{2}}^{J_{2}})}{m(C_{\underline{\sigma}_{1}}^{J_{1}})\,m(C_{\underline{\sigma}_{2}}^{J_{2}})}\leq
1+K_{0}\lambda_{0}^{-(n^{\prime}_{1}-n_{2})},$

with $\lambda_{0}:=\lambda/|\lambda_{2}|$, where $\lambda_{2}$ is the second
largest eigenvalue of $T$ in absolute value, and $K_{0}$ is a suitable
constant independent of $n_{1},n_{2},n_{1}^{\prime},n_{2}^{\prime}$.

###### Proof.

Calling $\\#(M,\sigma,\sigma^{\prime})$ the number of $T$-compatible sequences
of length $M+1$ starting with $\sigma$ end ending with $\sigma^{\prime}$, i.e.
such that

  * •

$\underline{\sigma}=(\underline{\sigma}(j_{1}),\ldots,\underline{\sigma}(j_{M+1}))$,

  * •

$T_{\underline{\sigma}(k)\underline{\sigma}(k+1)}=1$ for $k=1,\ldots,M$,

  * •

$\underline{\sigma}(j_{1})=\sigma$ and
$\underline{\sigma}(j_{M+1})=\sigma^{\prime}$,

then (A.1) gives

$\displaystyle\hbox{}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m(C_{\underline{\sigma}_{1}}^{J_{1}})$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\lim_{M\to\infty}\\!\frac{\\#(M-n_{1},1,\underline{\sigma}_{1}(n_{1}))\,\\#(M-n_{2},\underline{\sigma}_{1}(n_{2}),1)}{\\#(2M,1,1)},$
$\displaystyle\hbox{}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m(C_{\underline{\sigma}_{2}}^{J_{2}})$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\lim_{M\to\infty}\\!\frac{\\#(M-n_{1}^{\prime},1,\underline{\sigma}_{2}(n_{1}^{\prime}))\,\\#(M-n_{2}^{\prime},\underline{\sigma}_{2}(n_{2}^{\prime}),1)}{\\#(2M,1,1)},$
(A.2)
$\displaystyle\hbox{}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!m(C_{\underline{\sigma}_{1}}^{J_{1}}\\!\cap\\!C_{\underline{\sigma}_{2}}^{J_{2}})$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$
$\displaystyle\lim_{M\to\infty}\\!\frac{\\#(M\\!-\\!n_{1},1,\underline{\sigma}_{1}(n_{1}))\\#(n^{\prime}_{1}\\!-\\!n_{2},\underline{\sigma}_{1}(n_{2}),\underline{\sigma}_{2}(n^{\prime}_{1}))\\#(M\\!-\\!n_{2}^{\prime},\underline{\sigma}_{2}(n^{\prime}_{2}),1)}{\\#(2M,1,1)},$

where the boundary conditions on the sites $\pm(M+1)$ have been fixed equal to
1\. Clearly one has

$\\#(M,\sigma,\sigma^{\prime})=(T^{M+1})_{\sigma,\sigma^{\prime}}\,.$ (A.3)

Moreover $T$ has $s$ eigenvalues
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{s}$, with
$\lambda_{1}=\lambda>|\lambda_{k}|$ for $k=2,\ldots,s$. If
$v_{1},\ldots,v_{s}$ are the eigenvectors of $T$ associated with the
eigenvalues $\lambda_{1},\ldots,\lambda_{s}$, respectively, and $V$ denotes
the matrix with entries $V_{ij}=(v_{i})_{j}$, then one has

$(T^{n})_{\sigma\sigma^{\prime}}=\sum_{i=1}^{s}\lambda_{i}^{n}(V^{-1})_{i\sigma}V_{i\sigma^{\prime}},$
(A.4)

Replacing (A.4) in (A.3), inserting (A.3) in (A.1) and using that
$\lambda_{0}>1$ delivers the thesis. ∎

######  Remark A.2.

_The eigenvalues of the compatibility matrix of any Anosov automorphism on
$\mathds{T}^{2}$ are $\lambda$, $\lambda^{-1}$, together with $0$’s and roots
of unity [51]. Therefore we have $\lambda_{0}\geq\lambda$ in Lemma A.1. _

We can now extend Lemma A.1 to all bounded measurable functions as follows.
Let ${\mathcal{F}}^{\leq k}$ be the $\sigma$-algebra generated by the cylinder
sets $C_{\underline{\sigma}}^{J}$ with $J\subset(-\infty,k]$ and
${\mathcal{F}}^{\geq k}$ be the $\sigma$-algebra generated by the cylinders
$C_{\underline{\sigma}}^{J}$ with $J\subset[k,\infty)$.

######  Corollary A.3.

Given $k,k^{\prime}\in\mathds{Z}$, with $k^{\prime}>k$, let $g_{-}$ be a
bounded ${\mathcal{F}}^{\leq k}$-measurable function and $g_{-}$ be a bounded
${\mathcal{F}}^{\geq k^{\prime}}$-measurable function. Then one has

$\left|\langle g_{-}g_{+}\rangle-\langle g_{-}\rangle\langle
g_{+}\rangle\right|\leq
K_{1}\lambda_{0}^{-(k^{\prime}-k)}\|g_{-}\|_{\infty}\|g_{+}\|_{\infty}\,.$
(A.5)

for a suitable constant $K_{1}\geq K_{0}$, with $K_{0}$ s in Lemma A.1.

###### Proof.

Assume first that $k^{\prime}-k>a$. Let $B$ be a cylinder in
${\mathcal{F}}^{\geq k^{\prime}}$ and let ${\mathcal{G}}_{-}$ be the smallest
subset of ${\mathcal{F}}^{\leq k}$ containing all the sets $A$ such that

$1-K_{0}\lambda_{0}^{-(n^{\prime}_{1}-n_{2})}\leq\frac{m(A\cap
B)}{m(A)\,m(B)}\leq 1+K_{0}\lambda_{0}^{-(n^{\prime}_{1}-n_{2})}.$ (A.6)

Observe now that all cylinders of ${\mathcal{F}}^{\leq k}$ are in $\cal
G_{-}$. If $C^{J_{1}}_{\underline{\sigma}_{1}}$ and
$C^{J_{2}}_{\underline{\sigma}_{2}}$ are two cylinders in ${\mathcal{F}}^{\leq
k}$, then $C^{J_{1}}_{\underline{\sigma}_{1}}\cap
C^{J_{2}}_{\underline{\sigma}_{2}}$ is also a cylinder, while
$C^{J_{1}}_{\underline{\sigma}_{1}}\cup C^{J_{2}}_{\underline{\sigma}_{2}}$
can be written as the union of all cylinders $C^{J_{1}\cup
J_{2}}_{\underline{\sigma}^{\prime}}$ such that $\underline{\sigma}^{\prime}$
coincides with $\underline{\sigma}_{1}$ on $J_{1}$ or with
$\underline{\sigma}_{2}$ on $J_{2}$. In other words, the union of cylinders
can be decomposed into a possibly much larger, disjoint union of cylinders.

Finally, if $A=\bigcup_{i=1}^{k}A_{k}$, where $A_{1},\ldots,A_{k}$ are
disjoint measurable sets and $B$ is a measurable set, then

$\frac{m(A\cap B)}{m(A)m(B)}=\frac{\displaystyle{\sum_{i=1}^{k}m(A_{i}\cap
B)}}{\displaystyle{\sum_{i=1}^{k}m(A_{i})m(B)}},$

so that

$\min_{i=1,\ldots,k}\frac{m(A_{i}\cap B)}{m(A_{i})\,m(B)}\leq\frac{m(A\cap
B)}{m(A)\,m(B)}\leq\max_{i=1,\ldots,k}\frac{m(A_{i}\cap
B)}{m(A_{i})\,m(B)}\,.$

It follows that $\cal G_{-}$ contains the algebra generated by the cylinders.

On the other hand, if $\\{A_{i}\\}_{i=0}^{\infty}$ is a decreasing sequence of
sets in $\cal G_{-}$, then continuity of the measure implies that also
$\bigcap_{i=0}^{\infty}A_{i}\in{\mathcal{G}}_{-}$. The analogous statement for
a increasing sequence follows similarly since $m$ is a finite measure. In
conclusion, $\cal G_{-}$ is a monotone class that contains the algebra
generated by the cylinders and thus it contains ${\mathcal{F}}^{\leq k}$.

A similar argument for $B$ shows that (A.6) holds for all
$A\in{\mathcal{F}}^{\leq k}$ and $B\in{\mathcal{F}}^{\geq k^{\prime}}$.

Now, let $g_{-}$ and $g_{+}$ be simple functions, that is

$g_{-}(\underline{\sigma})=\sum_{i}a_{i}\chi_{A_{i}}^{J_{i}}(\underline{\sigma}),\qquad
g_{+}(\underline{\sigma})=\sum_{i^{\prime}}b_{i^{\prime}}\chi_{B_{i^{\prime}}}(\underline{\sigma}),$

where $\chi_{A}$ is the characteristic function of the set $A$, the sets
$A_{i}$ are disjoint ${\mathcal{F}}^{\leq k}$-measurable sets and the sets
$B_{i^{\prime}}$ are disjoint ${\mathcal{F}}^{\geq k^{\prime}}$-measurable
sets. We get

$\displaystyle\left|\langle g_{-}g_{+}\rangle-\langle g_{-}\rangle\langle
g_{+}\rangle\right|$
$\displaystyle=\sum_{i,i^{\prime}}|a_{k}||b_{i^{\prime}}|m(A_{i}\cap
B_{i^{\prime}})-m(A_{i})m(B_{i^{\prime}})|$ $\displaystyle\leq
K_{0}\lambda_{0}^{-(k^{\prime}-k)}\sum_{i,i^{\prime}}|a_{i}||b_{i^{\prime}}|m(A_{i})m(B_{i^{\prime}})\leq
K_{0}\lambda_{0}^{-(k^{\prime}-k)}\max_{i}|a_{i}|\max_{i^{\prime}}|b_{i^{\prime}}|\,.$

We can approximate any bounded ${\mathcal{F}}^{\leq k}$-measurable function
$g_{-}$ with a sequence $g_{-,k}$ of simple functions with
$\|g_{-,k}\|_{\infty}\leq\|g_{-}\|_{\infty}$ and $g_{-,k}\to g_{-}$ pointwise
$m$-almost everywhere, and similarly for $g_{+}$. We thus find that (A.5)
holds for every $g_{-}$ bounded ${\mathcal{F}}^{\leq k}$-measurable function
and $g_{+}$ bounded ${\mathcal{F}}^{\geq k^{\prime}}$-measurable function.
Finally we can remove the condition $k^{\prime}-k>a$ by replacing $K_{0}$ with
a suitable constant $K_{1}\geq K_{0}$. ∎

######  Remark A.4.

_If $g_{-}(\underline{\sigma})$ is ${\mathcal{F}}^{\leq k}$-measurable, then
$g_{-}(\underline{\sigma})$ depends only on $\sigma_{i}$ with $i\leq k$, that
is $g_{-}(\underline{\sigma})=g_{-}(\underline{\sigma}^{\prime})$ if
$\sigma_{i}=\sigma^{\prime}_{i}$ for every $i\leq k$. Similarly, if
$g_{+}(\underline{\sigma})$ is ${\mathcal{F}}^{\geq k}$, then
$g_{+}(\underline{\sigma})$ depends only on $\sigma_{i}$ with $i\geq
k^{\prime}$, that is
$g_{+}(\underline{\sigma})=g_{+}(\underline{\sigma}^{\prime})$ if
$\sigma_{i}=\sigma^{\prime}_{i}$ for every $i\geq k^{\prime}$._

###  A.2 Correlation functions

Let the sequences
$\underline{\sigma},\underline{\sigma}^{\prime}\in{\mathcal{N}}^{\mathds{Z}}_{T}$
be such that $\sigma_{i}=\sigma^{\prime}_{i}$ for $i\leq n$. If $n>0$ then
$d(\underline{\sigma},\underline{\sigma}^{\prime})\leq\lambda^{-n}$ and, for
$N\geq 0$,
$d(\tau^{-N}\underline{\sigma},\tau^{-N}\underline{\sigma}^{\prime})=\lambda^{-N}d(\underline{\sigma},\underline{\sigma}^{\prime})$.
Thus $\psi(\underline{\sigma}^{\prime})$ is on the unstable manifold of
$\psi(\underline{\sigma})$ and, since the unstable manifold is a straight
line, we have
$\psi(\underline{\sigma}^{\prime})=\psi(\underline{\sigma})+x\,v_{+}$, with
$|x|\leq C_{s}\lambda^{-n}$. If $n\leq 0$ we still have that
$d(\tau^{-N}(\underline{\sigma}),\tau^{-N}(\underline{\sigma}^{\prime}))\to 0$
as $N\to\infty$, so that even in this case we have
$\psi(\underline{\sigma}^{\prime})=\psi(\underline{\sigma})+xv_{+}$ for some
$x\in\mathds{R}$. Observe that
$\psi(\tau^{2n}\underline{\sigma}^{\prime})=\psi(\tau^{2n}\underline{\sigma})+\lambda^{2n}x\,v_{+}$
while, since
$(\tau^{2n}\underline{\sigma}^{\prime})_{i}=(\tau^{2n}\underline{\sigma})_{i}$
for $i\leq-n$, we have
$\|\psi(\tau^{2n}\underline{\sigma}^{\prime})-\psi(\tau^{2n}\underline{\sigma})\|\leq
C_{s}\lambda^{-n}$, so that, by the previous argument, we find again that
$|x|\leq C_{s}\lambda^{-n}$. In the same way one shows that, if
$\underline{\sigma}$ and $\underline{\sigma}^{\prime}$ are such that
$\sigma_{i}=\sigma^{\prime}_{i}$ for $i\geq n$, then
$\psi(\underline{\sigma}^{\prime})=\psi(\underline{\sigma})+xv_{-}$, with
$|x|\leq C_{s}\lambda^{n}$.

For
$\underline{\sigma},\underline{\sigma}^{\prime}\in{\mathcal{N}}_{T}^{\mathds{Z}}$
let
$\underline{\sigma}_{(-\infty,n]}\vee\underline{\sigma}_{(n,\infty)}^{\prime}$
be the sequence that agrees with $\underline{\sigma}$ on $(-\infty,n]$ and
with $\underline{\sigma}^{\prime}$ outside such an interval and call $m({\rm
d}\underline{\sigma}^{\prime}_{(n,\infty)}|\underline{\sigma}_{(-\infty,n]})$
the conditional probability measure on
$\underline{\sigma}^{\prime}_{(n,\infty)}$ given
$\underline{\sigma}_{(-\infty,n]}$ [29, prop. 5.3.2]. Calling
${\mathcal{N}}(\underline{\sigma})=\\{\underline{\sigma}^{\prime}\,|\,\underline{\sigma}_{(-\infty,n]}\vee\underline{\sigma}_{(n,\infty)}^{\prime}\in{\mathcal{N}}_{T}^{\mathds{Z}}\\}$
we have
$m({\mathcal{N}}(\underline{\sigma})|\underline{\sigma}_{(-\infty,n]})=1$.

Given $g\in{\mathfrak{B}}_{\alpha_{-},\alpha_{+}}(\mathds{T}^{2},\mathds{R})$
write $\hat{g}(\underline{\sigma}):=g(\psi(\underline{\sigma}))$ and observe
that
$\hat{g}(\underline{\sigma}_{(-\infty,n]}\vee\underline{\sigma}^{\prime}_{(n,\infty)})$
is almost everywhere well defined with respect to $m({\rm
d}\underline{\sigma}^{\prime}_{(n,\infty)}|\underline{\sigma}_{(-\infty,n]})$.
We can thus define

$\hat{g}^{(\leq
n)}(\underline{\sigma}):=\int\hat{g}(\underline{\sigma}_{(-\infty,n]}\vee\underline{\sigma}^{\prime}_{(n,\infty)})\,m({\rm
d}\underline{\sigma}^{\prime}_{(n,\infty)}|\underline{\sigma}_{(-\infty,n]}),$

By construction $\hat{g}^{(\leq n)}(\underline{\sigma})$ is
${\mathcal{F}}^{(\leq n)}$-measurable and $\langle\hat{g}^{(\leq
n)}\rangle=\langle\hat{g}\rangle$, where, with a slight abuse of notation, we
use $\langle\cdot\rangle$ also for the average with respect to $m$.
Analogously we define

$\hat{g}^{(\geq
n)}(\underline{\sigma}):=\int\hat{g}(\underline{\sigma}_{(-\infty,n)}^{\prime}\vee\underline{\sigma}_{[n,\infty)})\,m({\rm
d}\underline{\sigma}_{(-\infty,n)}^{\prime}|\underline{\sigma}_{[n,\infty)}),$

and considerations similar to those above still hold. Observe that
$\hat{g}^{(\leq n)}(\underline{\sigma})$ is the average of $g(\psi)$ over a
segment of the unstable manifold of $\psi(\underline{\sigma})$ of length
$O(\lambda^{-n})$. Thus if $n\gg 1$ we expect $\hat{g}^{(\leq
n)}(\underline{\sigma})$ to be very close to $\hat{g}(\underline{\sigma})$.

Finally, define

$\displaystyle\hat{g}^{(n,+)}(\underline{\sigma})$
$\displaystyle:=\hat{g}^{(\leq\lfloor
n/\alpha_{+}\rfloor)}(\underline{\sigma})-g^{(\leq\lfloor(n-1)/\alpha_{+}\rfloor)}(\underline{\sigma}),$
(A.7) $\displaystyle\hat{g}^{(n,-)}(\underline{\sigma})$
$\displaystyle:=\hat{g}^{(\geq\lfloor
n/\alpha_{-}\rfloor)}(\underline{\sigma})-g^{(\geq\lfloor(n+1)/\alpha_{-}\rfloor))}(\underline{\sigma})\,.$

From the argument above it follows that if $g$ is Hölder continuous along the
unstable manifold we expect $\hat{g}^{(n,+)}$ and $\hat{g}^{(-n,-)}$ to be
small for $n\gg 1$.

######  Remark A.5.

_Definition A.7 allows us to decompose_

$\hat{g}(\underline{\sigma})=\hat{g}^{(\leq
0)}(\underline{\sigma})+\sum_{k=1}^{\infty}\hat{g}^{(k,+)}(\underline{\sigma}),$

_where $\hat{g}^{(k,+)}(\underline{\sigma})$ depends only on
$\underline{\sigma}_{(-\infty,\lfloor k/\alpha_{+}\rfloor)}$. _

The above discussion is made precise in the following lemma.

######  Lemma A.6.

If $g$ is $(\alpha_{-},\alpha_{+})$-Hölder continuous on $\mathds{T}^{2}$,
then one has

$\displaystyle|\hat{g}^{(\leq
n)}(\underline{\sigma})-g(\underline{\sigma})|\leq
K_{2}\lambda^{-\alpha_{+}n}|g|_{\alpha_{+}}^{+}\,,\qquad|\hat{g}^{(\geq
n)}(\underline{\sigma})-g(\underline{\sigma})|\leq
K_{2}\lambda^{\alpha_{-}n}|g|_{\alpha_{-}}^{-},$ (A.8a)
$\displaystyle|\hat{g}^{(n,+)}(\underline{\sigma})|\leq
K_{2}\lambda^{-n}|g|_{\alpha_{+}}^{+},\qquad\qquad\qquad|\hat{g}^{(n,-)}(\underline{\sigma})|\leq
K_{2}\lambda^{n}|g|_{\alpha_{-}}^{-},$ (A.8b)

for a suitable positive constant $K_{2}$.

###### Proof.

If $\underline{\sigma}$ and $\underline{\sigma}^{\prime}$ are $T$-compatible
sequences and
$\underline{\sigma}_{(-\infty,n]}\vee\underline{\sigma}^{\prime}_{(n,\infty)}$
is $T$-compatible as well, then
$(\underline{\sigma})_{i}=(\underline{\sigma}_{(-\infty,n]}\vee\underline{\sigma}^{\prime}_{(n,\infty)})_{i}$
for $i\leq n$ and, from the discussion at the beginning of this subsection, we
get
$\psi(\underline{\sigma}_{(-\infty,n]}\vee\underline{\sigma}^{\prime}_{(n,\infty)})=\psi(\sigma)+xv_{+}$
with $|x|\leq C_{s}\lambda^{-n}$, so that

$|\hat{g}(\underline{\sigma}_{(-\infty,n]}\vee\underline{\sigma}^{\prime}_{(n,\infty)})-\hat{g}(\underline{\sigma})|\leq
C_{s}\lambda^{-n\alpha_{+}}|g|_{\alpha_{+}}^{+}\,.$

Integrating over $m({\rm
d}\underline{\sigma}^{\prime}_{(n,\infty)}|\underline{\sigma}_{(-\infty,n]})$
gives the first inequality in (A.8a). The second inequality can be derived in
a similar way.

Let now $\underline{\omega}$ be a $T$-compatible sequence and, for any pair of
symbols $\sigma,\sigma^{\prime}\in\\{1,\ldots,s\\}$, let
$\pi=\pi(\sigma,\sigma^{\prime})$ be a $T$-compatible sequence of length $a$
such that $T_{\sigma\pi_{1}}=T_{\pi_{a}\sigma^{\prime}}=1$. For every
$T$-compatible sequence $\underline{\sigma}$, define the $T$-compatible
sequence
$\underline{\omega}_{n}(\underline{\sigma}):=\underline{\sigma}_{(-\infty,n]}\vee\pi(\sigma_{n},\omega_{n+a})\vee\underline{\omega}_{[n+a,\infty)}$.
Reasoning as above we get

$\displaystyle|\hat{g}(\underline{\sigma}_{(-\infty,\lfloor(n-1)/\alpha_{+}\rfloor]}\vee\underline{\sigma}^{\prime}_{(\lfloor(n-1)/\alpha_{+}\rfloor,\infty)})-\hat{g}(\underline{\omega}_{\lfloor(n-1)/\alpha_{+}\rfloor}(\underline{\sigma}))|$
$\displaystyle\leq
C_{s}\lambda^{-\lfloor(n-1)/\alpha_{+}\rfloor\alpha_{+}}|g|_{\alpha_{+}}^{+}$
$\displaystyle\leq
C_{s}\lambda^{\alpha_{+}}\lambda^{-(n-1)}|g|_{\alpha_{+}}^{+}$

and, analogously,

$|\hat{g}(\underline{\sigma}_{(-\infty,\lfloor
n/\alpha_{+}\rfloor]}\vee\underline{\sigma}^{\prime}_{(\lfloor
n/\alpha_{+}\rfloor],\infty)})-\hat{g}(\underline{\omega}_{\lfloor(n-1)/\alpha_{+}\rfloor}(\underline{\sigma}))|\leq
C_{s}\lambda^{\alpha_{+}}\lambda^{-(n-1)}|g|_{\alpha_{+}}^{+}\,.$

Integrating over $m({\rm
d}\underline{\sigma}^{\prime}_{(\lfloor(n-1)/\alpha_{+}\rfloor,\infty)}|\underline{\sigma}_{(-\infty,\lfloor(n-1)/\alpha_{+}\rfloor]})$
and $m({\rm d}\underline{\sigma}^{\prime}_{(\lfloor
n/\alpha_{+}\rfloor,\infty)}|\underline{\sigma}_{(-\infty,\lfloor
n/\alpha_{+}\rfloor]})$, respectively, we get

$\displaystyle|\hat{g}^{(\leq\lfloor(n-1)/\alpha_{+}\rfloor)}(\underline{\sigma})-\hat{g}(\underline{\omega}_{\lfloor(n-1)/\alpha_{+}\rfloor}(\underline{\sigma}))|$
$\displaystyle\leq
C_{s}\lambda^{\alpha_{+}}\lambda^{-(n-1)}|g|_{\alpha_{+}}^{+},$
$\displaystyle|\hat{g}^{(\leq\lfloor
n/\alpha_{+}\rfloor)}(\underline{\sigma})-\hat{g}(\underline{\omega}_{\lfloor(n-1)/\alpha_{+}\rfloor}(\underline{\sigma}))|$
$\displaystyle\leq
C_{s}\lambda^{\alpha_{+}}\lambda^{-(n-1)}|g|_{\alpha_{+}}^{+},$

from which the first inequality of (A.8b) follows with
$K_{2}=2C_{s}\lambda^{1+\alpha_{+}}$. The second inequality is obtained in a
similar way. ∎

###  A.3 A correlation inequality: proof of Proposition 2.3

Since $\left\langle g_{+}\,g_{-}{\circ}A_{0}^{n}\right\rangle-\langle
g\rangle_{+}\langle
g\rangle_{-}=\left\langle\tilde{g}_{+}\,\tilde{g}_{-}{\circ}A_{0}^{n}\right\rangle$,
with the notations of (2.1), we can assume that $\langle g_{+}\rangle=\langle
g_{-}\rangle=0$. Then, calling $\tilde{n}:=\lfloor\alpha n\rfloor$, we write

$\displaystyle\hat{g}_{+}(\underline{\sigma})$
$\displaystyle=\hat{g}_{+}^{(\leq
0)}(\underline{\sigma})+\sum_{k=1}^{\tilde{n}}\hat{g}_{+}^{(k,+)}(\underline{\sigma})+(\hat{g}_{+}(\underline{\sigma})-\hat{g}_{+}^{(\leq\lfloor\tilde{n}/\alpha\rfloor)}(\underline{\sigma}))=:\sum_{k=0}^{\tilde{n}+1}\check{g}_{+}^{(k)}(\underline{\sigma}),$
$\displaystyle\hat{g}_{-}(\underline{\sigma})$
$\displaystyle=\hat{g}_{-}^{(\geq
0)}(\underline{\sigma})+\sum_{k^{\prime}=1}^{\tilde{n}}\hat{g}_{-}^{(-k^{\prime},-)}(\underline{\sigma})+(\hat{g}_{-}(\underline{\sigma})-\hat{g}_{-}^{(\geq-\lfloor\tilde{n}/\alpha\rfloor}(\underline{\sigma}))=:\sum_{k^{\prime}=0}^{\tilde{n}+1}\check{g}_{-}^{(k^{\prime})}(\underline{\sigma}),$

with obvious meaning of the symbols. By Lemma A.6 we get
$\|\check{g}_{+}^{(k)}\|_{\infty}\leq K_{2}\lambda^{-k}\|g_{+}\|_{\alpha}^{+}$
and, analogously, $\|\check{g}_{-}^{(k)}\|_{\infty}\leq
K_{2}\lambda^{-k}\|g_{-}\|_{\alpha}^{-}$.

Observe that
$(\tau^{n}\underline{\sigma})_{(-k^{\prime},\infty)}=\underline{\sigma}_{(n-k^{\prime},\infty)}$,
so that $\check{g}^{(k^{\prime})}_{-}(\tau^{n}\underline{\sigma})$ depends
only on $\underline{\sigma}_{(n-k^{\prime},\infty)}$. Thus if
$k+k^{\prime}<\tilde{n}$, by Corollary A.3 and Lemma A.6 we have

$\left|\left\langle\check{g}_{+}^{(k)}\,\check{g}_{-}^{(k^{\prime})}{\circ}\tau^{n}\right\rangle\right|\leq
K_{1}K_{2}\|g_{+}\|_{\alpha}^{+}\|g_{-}\|_{\alpha}^{-}\lambda^{-(k+k^{\prime})}\lambda_{0}^{-(n-\lfloor
k/\alpha\rfloor-\lfloor k^{\prime}/\alpha\rfloor)},$

while for $k+k^{\prime}\geq\tilde{n}$ we have, by Lemma A.6,

$\left|\left\langle\check{g}_{+}^{(k)}\,\check{g}_{-}^{(k^{\prime})}{\circ}\tau^{n}\right\rangle\right|\leq
K_{2}\|g_{+}\|_{\alpha}^{+}\|g_{-}\|_{\alpha}^{-}\lambda^{-(k+k)}\,.$

Summing over $k$ and $k^{\prime}$ we get

$|\left\langle g_{+}\,g_{-}{\circ}A_{0}^{n}\right\rangle|\leq
K_{2}\lambda^{-\tilde{n}}\|g_{+}\|_{\alpha}^{+}\|g_{-}\|_{\alpha}^{-}\left(\rule{0.0pt}{25.6073pt}\right.\sum_{\begin{subarray}{c}k,k^{\prime}=0,\ldots,\tilde{n}+1\\\
k+k^{\prime}<\tilde{n}\end{subarray}}K_{1}\lambda^{3}\lambda^{-(\tilde{n}-k-k^{\prime})\frac{1-\alpha}{\alpha}}+\sum_{\begin{subarray}{c}k,k^{\prime}=0,\ldots,\tilde{n}+1\\\
k+k^{\prime}\geq\tilde{n}\end{subarray}}\lambda^{\tilde{n}-k-k^{\prime}}\left)\rule{0.0pt}{25.6073pt}\right.,$

where we have used that $(n-\lfloor k/\alpha\rfloor-\lfloor
k^{\prime}/\alpha\rfloor)\geq(\tilde{n}-k-k^{\prime})/\alpha-3$, so that

$(k+k^{\prime})+(n-\lfloor k/\alpha\rfloor-\lfloor
k^{\prime}/\alpha\rfloor)\geq\tilde{n}/\alpha-(k+k^{\prime})(1-1/\alpha)-3\geq\tilde{n}+\frac{1-\alpha}{\alpha}(\tilde{n}-k-k^{\prime})-3\,.$

Thus we obtain

$|\left\langle
g_{+}\,g_{-}{\circ}A_{0}^{n}\right\rangle|\leq\max\\{1,K_{1}\\}K_{2}\lambda^{3}\|g_{+}\|_{\alpha}^{+}\|g_{-}\|_{\alpha}^{-}\lambda^{-\tilde{n}}\left(\tilde{n}\sum_{q=0}^{\tilde{n}}\lambda^{-q\frac{1-\alpha}{\alpha}}+\sum_{q=0}^{\tilde{n}+2}(\tilde{n}+q+1)\lambda^{-q}\right)$

from which the thesis follows observing that
$\lambda^{-\tilde{n}}\leq\lambda^{\alpha}\lambda^{-\alpha
n}\leq\lambda\,\lambda^{-\alpha n}$.

##  Appendix B Bounds on the norms of the iterated products

The following lemma is easily checked by direct computation.

######  Lemma B.1.

Let ${\mathscr{S}}$ be a map on ${\mathcal{U}}\times\mathds{T}^{2}$ of the
form (7.28), with ${\mathscr{S}}_{\varphi}(\varphi,\psi)$ of class $C^{3}$ in
$\varphi$, and let ${\mathfrak{p}}_{0},\ldots,{\mathfrak{p}}_{n-1}$ be any
functions defined in ${\mathcal{U}}\times\mathds{T}^{2}$ of class $C^{3}$ in
the first variable $\varphi$. Set
${\mathfrak{p}}^{(n)}(\varphi,\psi):={\mathfrak{p}}^{(n)}_{0}({\mathscr{S}};\,\psi)$,
with ${\mathfrak{p}}^{(n)}_{0}({\mathscr{S}};\varphi,\psi)$ defined in (7.29).
Then one has

$\displaystyle\partial_{\varphi}{\mathfrak{p}}^{(n)}$
$\displaystyle=\sum_{i=0}^{n-1}\bigl{(}\partial_{\varphi}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}\bigl{(}\partial_{\varphi}({\mathscr{S}}^{i})_{\varphi}\bigr{)}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\bigl{(}{\mathfrak{p}}_{j}{\circ}{\mathscr{S}}^{j}\bigr{)},$
(B.1a) $\displaystyle\partial^{2}_{\varphi}{\mathfrak{p}}^{(n)}$
$\displaystyle=\sum_{i=0}^{n-1}\Bigl{(}\\!\bigl{(}\partial_{\varphi}^{2}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}\bigl{(}\partial_{\varphi}({\mathscr{S}}^{i})_{\varphi}\bigr{)}^{2}+\bigl{(}\partial_{\varphi}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}\bigl{(}\partial_{\varphi}^{2}({\mathscr{S}}^{i})_{\varphi}\bigr{)}\Bigr{)}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\bigl{(}{\mathfrak{p}}_{j}{\circ}{\mathscr{S}}^{j}\bigr{)}$
(B.1b) $\displaystyle+\sum_{\begin{subarray}{c}i,j=0\\\
i\not=j\end{subarray}}^{n-1}\left(\partial_{\varphi}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\right)\left(\partial_{\varphi}({\mathscr{S}}^{i})_{\varphi}\right)\left(\partial_{\varphi}{\mathfrak{p}}_{j}{\circ}{\mathscr{S}}^{j}\right)\left(\partial_{\varphi}({\mathscr{S}}^{j})_{\varphi}\right)\prod_{\begin{subarray}{c}k=0\\\
k\not=i,j\end{subarray}}^{n-1}\bigl{(}{\mathfrak{p}}_{k}{\circ}{\mathscr{S}}^{k}\bigr{)},$
$\displaystyle\partial^{3}_{\varphi}{\mathfrak{p}}^{(n)}$
$\displaystyle=\sum_{i=0}^{n-1}\Bigl{(}\\!\bigl{(}\partial_{\varphi}^{3}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}\bigl{(}\partial_{\varphi}({\mathscr{S}}^{i})_{\varphi}\bigr{)}^{3}+3\bigl{(}\partial_{\varphi}^{2}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}\bigl{(}\partial_{\varphi}({\mathscr{S}}^{i})_{\varphi}\bigr{)}\bigl{(}\partial_{\varphi}^{2}({\mathscr{S}}^{i})_{\varphi}\bigr{)}$
(B.1c)
$\displaystyle+\bigl{(}\partial_{\varphi}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}\bigl{(}\partial_{\varphi}^{3}({\mathscr{S}}^{i})_{\varphi}\bigr{)}\\!\Bigr{)}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\bigl{(}{\mathfrak{p}}_{j}{\circ}{\mathscr{S}}^{j}\bigr{)}+2\sum_{\begin{subarray}{c}i,j=0\\\
i\not=j\end{subarray}}^{n-1}\Bigl{(}\\!\bigl{(}\partial_{\varphi}^{2}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}\bigl{(}\partial_{\varphi}({\mathscr{S}}^{i})_{\varphi}\bigr{)}^{2}$
$\displaystyle+\bigl{(}\partial_{\varphi}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}\bigl{(}\partial_{\varphi}^{2}({\mathscr{S}}^{i})_{\varphi}\bigr{)}\\!\Bigr{)}\left(\partial_{\varphi}{\mathfrak{p}}_{j}{\circ}{\mathscr{S}}^{j}\right)\bigl{(}\partial_{\varphi}({\mathscr{S}}^{j})_{\varphi}\bigr{)}\\!\\!\prod_{\begin{subarray}{c}k=0\\\
k\not=i,j\end{subarray}}^{n-1}\bigl{(}{\mathfrak{p}}_{k}{\circ}{\mathscr{S}}^{k}\bigr{)}+\sum_{\begin{subarray}{c}i,j,k=0\\\
i\not=j\neq k\neq
i\end{subarray}}^{n-1}\bigl{(}\partial_{\varphi}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}\bigr{)}$
$\displaystyle\times\bigl{(}\partial_{\varphi}({\mathscr{S}}^{i})_{\varphi}\bigr{)}\bigl{(}\partial_{\varphi}{\mathfrak{p}}_{j}{\circ}{\mathscr{S}}^{j}\bigr{)}\bigl{(}\partial_{\varphi}({\mathscr{S}}^{j})_{\varphi}\bigr{)}\bigl{(}\partial_{\varphi}{\mathfrak{p}}_{k}{\circ}{\mathscr{S}}^{k}\bigr{)}\bigl{(}(\partial_{\varphi}({\mathscr{S}}^{k})_{\varphi}\bigr{)}\prod_{\begin{subarray}{c}h=0\\\
h\not=i,j,k\end{subarray}}^{n-1}\bigl{(}{\mathfrak{p}}_{h}{\circ}{\mathscr{S}}^{h}\bigr{)}.$

######  Remark B.2.

_If both ${\mathscr{S}}_{\varphi}(\varphi,\psi)$ and the functions
${\mathfrak{p}}_{0},\ldots,{\mathfrak{p}}_{n-1}$ do not depend on $\psi$, then
setting
${\mathfrak{p}}^{(n)}(\varphi)={\mathfrak{p}}^{(n)}_{0}({\mathscr{S}};\varphi)$,
equations (B.1) still hold, with ${\mathscr{S}}(\varphi,\psi)$ and
${\mathfrak{p}}_{i}({\mathscr{S}}^{i}(\varphi\psi))$ replaced, respectively,
with $G(\varphi)$ and ${\mathfrak{p}}_{i}(G^{i}(\varphi))$. _

###  B.1 A first application: completion of the proof of Theorem 2

Differentiating twice the function $h_{1}$ in (6.24) gives

$\|\partial_{\theta}^{2}h_{1}\|_{\infty}\leq\sum_{n=1}^{\infty}\left(\bigl{\|}\partial_{\theta}^{2}p_{1}^{(n)}\bigr{\|}_{\infty}\|q_{1}\|_{\infty}+\bigl{\|}\partial_{\theta}p_{1}^{(n)}\bigr{\|}_{\infty}\|\partial_{\theta}(q_{1}{\circ}{\mathscr{S}}_{1}^{n})\|_{\infty}+\bigl{\|}p_{1}^{(n)}\bigr{\|}_{\infty}\|\partial_{\theta}^{2}(q_{1}{\circ}{\mathscr{S}}_{1}^{n})\|_{\infty}\right),$

where $\bigl{\|}\partial_{\theta}p_{1}^{(n)}\bigr{\|}_{\infty}$ can be bounded
as in 6.36. Thus, noting that

$\partial_{\theta}(q_{1}{\circ}{\mathscr{S}}_{1}^{n})=(\partial_{\theta}q_{1}{\circ}{\mathscr{S}}_{1}^{n})\,\partial_{\theta}{\mathscr{S}}_{1}^{n},\qquad\partial_{\theta}^{2}(q_{1}{\circ}{\mathscr{S}}_{1}^{n})=(\partial_{\theta}^{2}q_{1}{\circ}{\mathscr{S}}_{1}^{n})\,(\partial_{\theta}{\mathscr{S}}_{1}^{n})^{2}+(\partial_{\theta}q_{1}{\circ}{\mathscr{S}}_{1}^{n})\,\partial_{\theta}^{2}{\mathscr{S}}_{1}^{n},$

and writing $\partial_{\theta}^{2}p_{1}^{(n)}$ according to (B.1b), we can use
Lemma 6.11 to obtain the bound (6.38), for a suitable constant $D_{3}$.

Finally, since

$\partial_{\theta}h_{1}=\sum_{n=1}^{\infty}\left(\bigl{(}\partial_{\theta}p_{1}^{(n)}\bigr{)}q_{1}+p_{1}^{(n)}\partial_{\theta}(q_{1}{\circ}{\mathscr{S}}_{1}^{n})\right),$

we find

$\displaystyle|\partial_{\theta}h_{1}|_{\alpha_{*}}$
$\displaystyle\leq\sum_{n=1}^{\infty}\Bigl{(}\bigl{|}\partial_{\theta}p_{1}^{(n)}\bigr{|}_{\alpha_{*}}\|q_{1}\|_{\infty}+\bigl{\|}\partial_{\theta}p_{1}^{(n)}\bigr{\|}_{\infty}|q_{1}{\circ}{\mathscr{S}}_{1}^{n}|_{\alpha_{*}}$
$\displaystyle\qquad\qquad+\bigl{|}p_{1}^{(n)}\bigr{|}_{\alpha_{*}}\|\partial_{\theta}q\|_{\infty}+\bigl{\|}p_{1}^{(n)}\bigr{\|}_{\infty}|\partial_{\theta}q_{1}{\circ}{\mathscr{S}}_{1}^{n}|_{\alpha_{*}}\Bigr{)},$

where the factors which are not differentiated with respect to $\theta$ are
bounded as in done to obtain (6.40), while the differentiated factors are
bounded once more by relaying on (B.1a) and the bounds in Lemma 6.11.

###  B.2 Products independent of the fast variable: proof of Lemma 7.16

Let $\overline{N}=O(\rho^{-1})$ be defined as in Remark 7.15. Then, using that
$|\overline{G}^{n}(\varphi)|\leq\overline{\theta}$ for $n\geq\overline{N}$ and
that

$\partial_{\varphi}\overline{G}^{n}(\varphi)=\prod_{i=0}^{n-1}\partial_{\varphi}\overline{G}(\overline{G}^{i}(\varphi)),\vspace{-.2cm}$
(B.2)

we get

$\|\partial_{\varphi}\overline{G}^{n}\|_{\infty}\leq
C(1-\rho\,\gamma^{\prime})^{n}\,.$ (B.3)

Then, combining (B.1a) and (7.30) together with the bound (B.3), and reasoning
like in (6.34) and (6.36), we obtain

$\displaystyle\left|\prod_{i=0}^{n-1}{\mathfrak{p}}_{i}(\overline{G}^{i}(\varphi)\right|$
$\displaystyle\leq C(1-\rho\,\gamma^{\prime})^{n}$ (B.4a)
$\displaystyle\left|\partial_{\varphi}\left(\prod_{i=0}^{n-1}{\mathfrak{p}}_{i}(\overline{G}^{i}(\varphi))\right)\right|$
$\displaystyle\leq
C\rho(1-\rho\,\gamma^{\prime})^{n-1}\sum_{i=0}^{n-1}(1-\rho\,\gamma^{\prime})^{i}\leq
C(1-\rho\,\gamma^{\prime})^{n},$ (B.4b)

so that (B.4a) gives

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|{\mathfrak{p}}^{(n)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,0}=\|{\mathfrak{p}}^{(n)}\|_{\infty}\leq
C(1-\rho\,\gamma^{\prime})^{n},$

while (B.4b) gives

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|{\mathfrak{p}}^{(n)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,1}\leq
C(1-\rho\,\gamma^{\prime})^{n},\qquad\|\partial_{\varphi}^{2}\overline{G}^{n}\|_{\infty}\leq
C(1-\rho\,\gamma^{\prime})^{n},$ (B.5)

where the second one is a special case of the first one, with
${\mathfrak{p}}_{i}=\partial_{\varphi}\overline{G}$ $\forall i=0,\ldots,n-1$.
Inserting the bounds (B.5) in (B.1b) (again see (7.30) for notations) and
reasoning like in (B.4b), we get

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|{\mathfrak{p}}^{(n)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}\leq
C(1-\rho\,\gamma^{\prime})^{n},\qquad\|\partial_{\varphi}^{3}\overline{G}^{n}\|_{\infty}\leq
C(1-\rho\,\gamma^{\prime})^{n}.$ (B.6)

Using (B.6) in (B.1c) delivers the inequality (7.36).

Finally, from (B.5), (B.6) and the bound (B.3) one obtains (7.37).

###  B.3 Products depending on the fast variable: proof of Lemma 7.27

Assume the hypotheses of Lemma 7.27 to be satisfied. Reasoning as in the proof
of Lemma 7.16, we find

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|{\mathfrak{p}}^{(n)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,3}\leq
C(1-\rho\,\gamma^{\prime})^{n}.$ (B.7)

Since
$\partial_{\theta}({\mathscr{S}}^{n}_{2})_{\theta}=(\partial_{\theta}G_{2})^{(n)}$,
where $(\partial_{\theta}G_{2})^{(n)}$ is given by (7.29) with
${\mathfrak{p}}_{i}=\partial_{\theta}G_{2}$ for all $i=0,\ldots,n-1$, and
$({\mathscr{S}}_{2})_{\theta}(0,\psi)=0$, we also have

$\|\partial^{k}_{\theta}({\mathscr{S}}^{n}_{2})_{\theta}\|_{\infty}\leq
C(1-\rho\,\gamma^{\prime})^{n},\qquad k=0,1,2,3.$ (B.8)

On the other hand writing

$\displaystyle G_{2}({\mathscr{S}}_{2}^{n}(\theta,\psi))$ $\displaystyle-
G_{2}({\mathscr{S}}_{2}^{n}(\theta,\psi^{\prime}))=G_{2}(({\mathscr{S}}_{2}^{n})_{\theta}(\theta,\psi),A_{0}^{n}\psi)-G_{2}(({\mathscr{S}}_{2}^{n})_{\theta}(\theta,\psi^{\prime}),A_{0}^{n}\psi)$
$\displaystyle+({\mathscr{S}}_{2}^{n})_{\theta}(\theta,\psi^{\prime})\int_{0}^{1}\left(\partial_{\theta}G_{2}(t({\mathscr{S}}_{2}^{n})_{\theta}(\theta,\psi^{\prime}),A_{0}^{n}\psi)-\partial_{\theta}G_{2}(t({\mathscr{S}}_{2}^{n})_{\theta}(\theta,\psi^{\prime}),A_{0}^{n}\psi^{\prime})\right){\rm
d}t$

we have, for $n\geq 2$,

$|({\mathscr{S}}_{2}^{n})_{\theta}|_{\alpha_{0}}^{-}=|G_{2}{\circ}{\mathscr{S}}^{n-1}_{2}|_{\alpha_{0}}^{-}\leq\|(\partial_{\theta}G_{2}){\circ}{\mathscr{S}}^{n-1}_{2}\|_{\infty}|G_{2}{\circ}{\mathscr{S}}^{n-2}_{2}|_{\alpha_{0}}^{-}+(1-\rho\,\gamma^{\prime})^{n-1}\lambda^{-\alpha_{0}(n-1)}|\partial_{\theta}G_{2}|^{-}_{\alpha_{0}},$

so that, iterating, thanks to the condition assumed on $\rho$, we get

$\displaystyle|({\mathscr{S}}_{2}^{n})_{\theta}|_{\alpha_{0}}^{-}$
$\displaystyle\leq|{\hskip-0.85355pt}|{\hskip-0.85355pt}|G_{2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},1}\sum_{i=0}^{n-1}(1-\rho\,\gamma^{\prime})^{n-1-i}\lambda^{-\alpha_{0}(n-1-i)}\prod_{j=1}^{i}\|(\partial_{\theta}G_{2}){\circ}{\mathscr{S}}^{n-j}_{2}\|_{\infty}$
(B.9) $\displaystyle\leq
C|{\hskip-0.85355pt}|{\hskip-0.85355pt}|G_{2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},1}(1-\rho\,\gamma^{\prime})^{n-1}\sum_{i=0}^{n-1}\lambda^{-\alpha_{0}(n-1-i)}\leq
C(1-\rho\,\gamma^{\prime})^{n}\,,$

which holds true also for $n=1$. We can now write, by (2.7),

$|{\mathfrak{p}}^{(n)}|_{\alpha_{0}}^{-}\leq\sum_{i=0}^{n-1}|{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}_{2}|_{\alpha_{0}}^{-}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\|{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{j}_{2}\|_{\infty},$

where

$|{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}_{2}|_{\alpha_{0}}^{-}\leq\|\partial_{\theta}{\mathfrak{p}}_{i}\|_{\infty}|({\mathscr{S}}^{i}_{2})_{\theta}|_{\alpha_{0}}^{-}+\lambda^{-\alpha_{0}i}|{\mathfrak{p}}_{i}|_{\alpha_{0}},$

with the first term missing if $i=0$, so that we get

$\displaystyle|{\mathfrak{p}}^{(n)}|_{\alpha_{0}}^{-}$ $\displaystyle\leq
C(1-\rho\,\gamma^{\prime})^{n-1}\left(\rho\sum_{i=1}^{n-1}(1-\rho\,\gamma^{\prime})^{i}+\rho\sum_{i=0}^{n-1}\lambda^{-\alpha_{0}i}\right)\leq
C(1-\rho\,\gamma^{\prime})^{n},$ (B.10)

which implies, taking ${\mathfrak{p}}_{i}=\partial_{\theta}G_{2}$ $\forall
i=0,\ldots,n-1$, also the bound

$|\partial_{\theta}({\mathscr{S}}^{n}_{2})_{\theta}|_{\alpha_{0}}^{-}\leq
C(1-\rho\,\gamma^{\prime})^{n}.$ (B.11)

Analogously, using the expression (B.1b) for
$\partial_{\theta}^{2}{\mathfrak{p}}^{(n)}(\theta)$ and (B.8), we get

$\displaystyle|\partial_{\theta}{\mathfrak{p}}^{(n)}|_{\alpha_{0}}^{-}\leq$
$\displaystyle\sum_{i=0}^{n-1}\Bigl{(}|\partial_{\theta}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}_{2}|_{\alpha_{0}}^{-}\|\partial_{\theta}({\mathscr{S}}^{i}_{2})_{\theta}\|_{\infty}+\|\partial_{\theta}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}_{2}\|_{\infty}|\partial_{\theta}({\mathscr{S}}^{i}_{2})_{\theta}|_{\alpha_{0}}^{-}\Bigr{)}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\|{\mathfrak{p}}_{j}{\circ}{\mathscr{S}}^{j}_{2}\|_{\infty}$
$\displaystyle+$ $\displaystyle\sum_{\begin{subarray}{c}i,j=0\\\ i\neq
j\end{subarray}}^{n-1}\|\partial_{\theta}{\mathfrak{p}}_{i}{\circ}{\mathscr{S}}^{i}_{2}\|_{\infty}\|\partial_{\theta}({\mathscr{S}}^{i}_{2})_{\theta}\|_{\infty}|{\mathfrak{p}}_{j}{\circ}{\mathscr{S}}^{j}_{2}|_{\alpha_{0}}^{-}\prod_{\begin{subarray}{c}k=0\\\
k\not=i,j\end{subarray}}^{n-1}\|{\mathfrak{p}}_{k}{\circ}{\mathscr{S}}^{k}_{2}\|_{\infty},$

that, by the same argument used in (B.10), delivers

$|\partial_{\theta}{\mathfrak{p}}^{(n)}|_{\alpha_{0}}^{-}\leq
C\,(1-\rho\,\gamma^{\prime})^{n},\qquad\qquad|\partial^{2}_{\theta}({\mathscr{S}}^{n}_{2})_{\theta}|_{\alpha_{0}}^{-}\leq
C\,(1-\rho\,\gamma^{\prime})^{n}.$ (B.12)

It is now easy, by using the expression (B.1c) for
$\partial_{\theta}^{3}p^{(n)}(\theta)$ and the bounds (B.8)–(B.12), and
reasoning once more as in in (B.10), to obtain

$|\partial_{\theta}^{2}{\mathfrak{p}}^{(n)}|_{\alpha_{0}}^{-}\leq
C\,(1-\rho\,\gamma^{\prime})^{n},$ (B.13)

so as to complete the proof of (7.53).

Finally the bounds (7.54) follow collecting together the bounds (B.8)–(B.12).

##  Appendix C A correlation inequality involving the slow variable

In this and the following Appendix D, the maps ${\mathscr{S}}$,
${\mathscr{S}}_{1}$ and ${\mathscr{S}}_{2}$ are meant as the extended maps
which are obtained by following the procedure described in Subsection 7.1,
and, analogously, the domains are meant as the extended domains where the
extended maps are defined.

###  C.1 Some preliminary rewriting

Define
${\mathfrak{p}}^{(k)}_{i}:={\mathfrak{p}}^{(k)}_{i}({\mathscr{S}}_{2};\theta,\psi)$
and
$\langle{\mathfrak{p}}\rangle^{(k)}_{i}:=\langle{\mathfrak{p}}\rangle^{(k)}_{i}(\overline{\mathscr{S}};\theta)$,
for $k=0,\ldots,n-1$ and $i=0,\ldots,n-k$. according to (7.29) and (7.31),
respectively, and set ${\mathfrak{p}}^{(n)}={\mathfrak{p}}^{(n)}_{0}$ and
$\langle{\mathfrak{p}}\rangle^{(n)}=\langle{\mathfrak{p}}\rangle^{(n)}_{0}$.
Observe that

$\displaystyle{\mathfrak{p}}^{(n)}(\theta,\psi)\,g_{-}({\mathscr{S}}_{2}^{n}(\theta,\psi))-\langle{\mathfrak{p}}\rangle^{(n)}(\theta)\,g_{-}(\overline{\mathscr{S}}^{n}(\theta,\psi))$
(C.1)
$\displaystyle\qquad\qquad=\sum_{i=0}^{n-1}\Bigl{(}\langle{\mathfrak{p}}\rangle^{(i)}(\theta)\,{\mathfrak{p}}_{i}^{(n-i)}(\overline{\mathscr{S}}^{i}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{n-i}(\overline{\mathscr{S}}^{i}(\theta,\psi)))$
$\displaystyle\qquad\qquad\qquad\qquad-\langle{\mathfrak{p}}\rangle^{(i+1)}(\theta)\,{\mathfrak{p}}_{i+1}^{(n-i-1)}(\overline{\mathscr{S}}^{i+1}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{n-i-1}(\overline{\mathscr{S}}^{i+1}(\theta,\psi)))\Bigr{)}.$

Moreover we have

${\mathfrak{p}}^{(n-i)}_{i}(\overline{\mathscr{S}}^{i}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{n-i}(\overline{\mathscr{S}}^{i}(\theta,\psi)))={\mathfrak{p}}^{(n-i)}_{i}(\overline{G}^{i}(\theta),A_{0}^{i}\psi)\,g_{-}({\mathscr{S}}_{2}^{n-i}(\overline{G}^{i}(\theta),A_{0}^{i}\psi)),$
(C.2)

that is
$\langle{\mathfrak{p}}\rangle^{(i)}(\theta)\,{\mathfrak{p}}_{i}^{(n-i)}(\overline{\mathscr{S}}^{i}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{n-i}(\overline{\mathscr{S}}^{i}(\theta,\psi)))$
depends on $\psi$ only through $\psi^{\prime}:=A_{0}^{i}\psi$; a similar
consideration holds for the second term inside the summation in (C.1).
Defining

$\Delta^{(k)}_{i}(\theta,\psi):={\mathfrak{p}}^{(k)}_{i}(\theta,\psi)\,g_{-}({\mathscr{S}}_{2}^{k}(\theta,\psi))-\langle{\mathfrak{p}}_{i}\rangle(\theta)\,{\mathfrak{p}}^{(k-1)}_{i+1}(\overline{G}(\theta),A_{0}\psi)\,g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{G}(\theta),A_{0}\psi)),$

we get

${\mathfrak{p}}^{(n)}(\theta,\psi)\,g_{-}({\mathscr{S}}_{2}^{n}(\theta,\psi))-\langle{\mathfrak{p}}\rangle^{(n)}(\theta)\,g_{-}(\overline{\mathscr{S}}^{n}(\theta,\psi))=\sum_{i=0}^{n-1}\langle{\mathfrak{p}}\rangle^{(i)}(\theta)\,\Delta^{(n-i)}_{i}(\overline{\mathscr{S}}^{i}(\theta,\psi)).$
(C.3)

###  C.2 The new correlation inequality: proof of Proposition 7.30

We start with a particular case, by assuming the function $g_{+}$ in
Proposition 7.30 to have zero average. Eventually we extend the result to any
$g_{+}\in{\mathscr{B}}_{\alpha_{0}}^{+}(\Omega,\mathds{R})$.

######  Lemma C.1.

Assume $\rho$ to be such that the map ${\mathscr{S}}_{2}$ satisfies Hypotheses
1–3. Let ${\mathfrak{p}}_{0},\ldots,{\mathfrak{p}}_{n-1}$ be any functions in
${\mathscr{B}}_{0,3}(\Omega,\mathds{R})$ such that, for some
$\rho^{\prime}\in(0,1)$,

  1. 1.

$\|{\mathfrak{p}}_{i}-1\|_{\infty}=O(\rho)$ for all $i=0,\ldots,n-1$,

  2. 2.

$|{\mathfrak{p}}_{i}(\theta)|\leq 1-\rho\gamma^{\prime}$ for
$|\theta|\leq\theta^{\prime}$ for some $\theta^{\prime}=O(1)$ in $\rho$ and
for all $i=0,\ldots,n-1$,

and set
${\mathfrak{p}}^{(n)}(\theta,\psi):={\mathfrak{p}}^{(n)}_{0}({\mathscr{S}}_{2};\theta,\psi)$,
with ${\mathfrak{p}}^{(n)}_{0}({\mathscr{S}}_{2};\theta,\psi)$ defined
according to (7.29). Then, for any
$g_{+}\in{\mathscr{B}}_{\alpha_{0}}^{+}(\Omega,\mathds{R})$with $\langle
g_{+}\rangle(\theta)\equiv 0$, and
$g_{-}\in{\mathscr{B}}_{\alpha_{0},2}^{-}(\Omega,\mathds{R})$, one has

$\left|\left\langle
g_{+}{\mathfrak{p}}^{(n)}g_{-}{\circ}{\mathscr{S}}_{2}^{n}\right\rangle\right|\leq
C(1-\rho\,\gamma^{\prime})^{n}\left(\lambda^{-\alpha_{0}n}\|g_{+}\|^{+}_{\alpha_{0}}\|\tilde{g}_{-}\|^{-}_{\alpha_{0}}+\rho\,\|g_{+}\|^{+}_{\alpha_{0}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|^{-}_{\alpha_{0},2}\right),$

where the constant $C$ does not depends on $n$.

###### Proof.

Note that
$\Delta^{(n-i)}_{i}(\overline{\mathscr{S}}^{i}(\theta,\psi))=\Delta^{(n-i)}_{i}(\overline{G}^{i}(\theta),A_{0}\psi))$
in (C.3) depends on $\psi$ only through $\psi^{\prime}:=A_{0}^{i}\psi$. Thus,
using Proposition 2.3, we bound

$\displaystyle\left|\left\langle
g_{+}{\mathfrak{p}}^{(n)}g_{-}{\circ}{\mathscr{S}}_{2}^{n}\right\rangle\right|$
$\displaystyle\leq\left|\left\langle
g_{+}{\mathfrak{p}}^{(n)}g_{-}{\circ}{\mathscr{S}}_{2}^{n}\right\rangle-\langle{\mathfrak{p}}\rangle^{(n)}\left\langle
g_{+}g_{-}{\circ}\overline{\mathscr{S}}^{n}\right\rangle\right|+\bigl{|}\langle{\mathfrak{p}}\rangle^{(n)}\bigr{|}\left|\left\langle
g_{+}\,g_{-}{\circ}\overline{\mathscr{S}}^{n}\right\rangle\right|$ (C.4)
$\displaystyle\leq
C\sum_{i=0}^{n-1}\|{\mathfrak{p}}^{(i)}\|_{\infty}(1+\alpha_{0}i)\,\lambda^{-\alpha_{0}i}\|g_{+}\|_{\alpha_{0}}^{+}\|\Delta^{(n-i)}_{i}\|_{\alpha_{0}}^{-}$
$\displaystyle+C\|\langle{\mathfrak{p}}\rangle^{(n)}\|_{\infty}(1+\alpha_{0}n)\,\lambda^{-\alpha_{0}n}\|g_{+}\|^{+}_{\alpha_{0}}\|\tilde{g}_{-}\|^{-}_{\alpha_{0}}.$

Observe finally that
${\mathscr{S}}_{2}(\theta,\psi)=(\overline{G}(\theta)+\rho\Delta
f(\theta,\psi),A_{0}\psi)$, with

$\Delta f(\theta,\psi):=\tilde{f}(\theta,\psi)-\chi(\theta)\,f(0,\psi),$ (C.5)

so that we have

$\displaystyle\Delta^{(k)}_{i}(\theta,\psi)$
$\displaystyle=\bigl{(}{\mathfrak{p}}_{i}(\theta,\psi)-\langle{\mathfrak{p}}_{i}\rangle(\theta)\bigr{)}{\mathfrak{p}}_{i+1}^{(k-1)}({\mathscr{S}}_{2}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k}(\theta,\psi))$
$\displaystyle+\langle{\mathfrak{p}}_{i}\rangle(\theta)\Bigl{(}{\mathfrak{p}}_{i+1}^{(k-1)}(\overline{G}(\theta)+\rho\Delta
f(\theta,\psi),A_{0}\psi)-{\mathfrak{p}}_{i+1}^{(k-1)}(\overline{G}(\theta),A_{0}\psi)\Bigr{)}g_{-}({\mathscr{S}}_{2}^{k}(\theta,\psi))$
$\displaystyle+\langle{\mathfrak{p}}_{i}\rangle(\theta)\,{\mathfrak{p}}_{i+1}^{(k-1)}(\overline{G}(\theta),A_{0}\psi)\bigl{(}g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{G}(\theta,\psi)+\rho\Delta
f(\theta,\psi))-g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{G}(\theta),A_{0}\psi))\bigr{)}.$

Thus, writing

$\displaystyle{\mathfrak{p}}_{i+1}^{(k-1)}(\overline{G}(\theta)+\rho\Delta
f(\theta,\psi),A_{0}\psi)-{\mathfrak{p}}_{i+1}^{(k-1)}(\overline{G}(\theta),A_{0}\psi)$
(C.6) $\displaystyle\qquad=\rho\Delta f(\theta,\psi)\int_{0}^{1}{\rm
d}t\,\partial_{\theta}{\mathfrak{p}}_{i+1}^{(k-1)}(\overline{G}(\theta)+t\rho\Delta
f(\theta,\psi),A_{0}\psi),$

and

$\displaystyle g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{G}(\theta)+\rho\Delta
f(\theta,\psi))-g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{G}(\theta),A_{0}\psi))=\rho\Delta
f(\theta,\psi)\int_{0}^{1}{\rm d}t$ (C.7)
$\displaystyle\quad\times\left(\left(\partial_{\theta}g_{-}{\circ}{\mathscr{S}}_{2}^{k-1}(\overline{G}(\theta)+t\rho\Delta
f(\theta,\psi),A_{0}\psi)\right)\left(\partial_{\theta}({\mathscr{S}}_{2}^{k-1})_{\theta}(\overline{G}(\theta)+t\rho\Delta
f(\theta,\psi),A_{0}\psi)\right)\right),$

and using that

$\bigl{|}\partial_{\theta}g_{-}{\circ}{\mathscr{S}}_{2}^{k}\bigr{|}_{\alpha_{0}}^{-}\leq\|\partial^{2}_{\theta}g_{-}{\circ}{\mathscr{S}}_{2}^{k}\|_{\infty}\bigl{|}({\mathscr{S}}_{2}^{k})_{\theta}\bigr{|}_{\alpha_{0}}^{-}+\lambda^{-\alpha_{0}k}\bigl{|}\partial_{\theta}g_{-}\bigr{|}_{\alpha_{0}}^{-},$

then the assumptions on the functions
${\mathfrak{p}}_{0},\ldots,{\mathfrak{p}}_{n-1}$ and the bounds provided in
Lemma 7.27 and Remark 7.28 give

$\|\Delta^{(k)}_{i}\|_{\alpha_{0}}^{-}\leq
C\rho(1-\rho\,\gamma^{\prime})^{k-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}^{-},$

which, inserted into (C.4), completes the proof. ∎

######  Remark C.2.

_Note that, using ( C.6) and (C.7) and still relying on the bounds of Lemma
7.27 and assuming $g_{-}$ to be in
${\mathscr{B}}_{\alpha_{0},3}^{-}(\Omega,\mathds{R})$, in the same way we get
the bound on $\|\Delta^{(k)}\|_{\alpha_{0}}^{-}$ we may prove more generally
the bound_

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\Delta^{(k)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}^{-}\leq
C\rho(1-\rho\,\gamma^{\prime})^{k-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}.$

######  Lemma C.3.

Assume $\rho$ to be such that the map ${\mathscr{S}}_{2}$ satisfies Hypotheses
1–3. Let ${\mathfrak{p}}_{0},\ldots,{\mathfrak{p}}_{n-1}$ be any functions in
${\mathscr{B}}_{0,3}(\Omega,\mathds{R})$ as in Lemma C.1, and let
$\langle{\mathfrak{p}}\rangle^{(n)}$ defined as in (7.29). Then, for any
$g_{-}\in{\mathscr{B}}_{\alpha_{0},2}^{-}(\Omega,\mathds{R})$, if one defines

$\Xi^{(k)}(\theta,\psi):=\langle{\mathfrak{p}}\rangle^{(k)}(\theta)\,g_{-}(\overline{\mathscr{S}}^{k}(\theta,\psi)),$

one has

$\left|\left\langle\Xi^{(k)}{\circ}{\mathscr{S}}_{2}-\Xi^{(k)}{\circ}\overline{\mathscr{S}}\right\rangle\right|\leq
C\rho(1-\rho\,\gamma^{\prime})^{k}\bigl{(}(1+\alpha_{0}k)\,\lambda^{-\alpha_{0}k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}^{-}+\rho|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}\bigr{)}\,.$

###### Proof.

Writing

$\displaystyle\Xi^{(k)}({\mathscr{S}}_{2}(\theta,\psi))-\Xi^{(k)}(\overline{\mathscr{S}}(\theta,\psi))$
$\displaystyle\quad=\rho\Delta
f(\theta,\psi)\,\partial_{\theta}\Xi^{(k)}(\overline{\mathscr{S}}(\theta,\psi))+(\rho\Delta
f(\theta,\psi))^{2}\\!\\!\int_{0}^{1}{\rm
d}t\,(1-t)\,\partial^{2}_{\theta}\Xi^{(k)}(\overline{G}(\theta,\psi)+t\rho\Delta
f(\theta,\psi),A_{0}\psi),$

with $\Delta f(\theta,\psi)$ as in (C.5), and observing that

$\Xi^{(k)}(\overline{\mathscr{S}}(\theta,\psi))=\langle{\mathfrak{p}}\rangle^{(k)}(\overline{G}(\theta))\,g_{-}(\overline{G}^{k+1}(\theta),A_{0}^{k+1}\psi)),$

we get, by using Lemmas 7.16 and C.1,

$\left|\left\langle\Delta
f\,\partial_{\theta}\Xi^{(k)}{\circ}\overline{\mathscr{S}}\right\rangle\right|\leq
C(1+\alpha_{0}k)\,\lambda^{-\alpha_{0}k}\|\Delta
f\|_{\alpha_{0}}(1-\rho\,\gamma^{\prime})^{k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}^{-},$

while, using (2.10), we find, again by Lemma 7.16,

$\|\partial^{2}_{\theta}\Xi^{(k)}\|_{\infty}\leq
C|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\langle{\mathfrak{p}}\rangle^{(k)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}\,|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}{\circ}\overline{\mathscr{S}}^{(k)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}\leq
C(1-\rho\,\gamma^{\prime})^{k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}\,.$

Combining the above estimates we obtain the desired bound. ∎

Now we can prove Proposition 7.30. Assume $\rho$ to be such that the map
${\mathscr{S}}_{2}$ satisfies Hypotheses 1–3, and observe that

$\left\langle
g_{+}{\mathfrak{p}}^{(n)}g_{-}{\circ}{\mathscr{S}}_{2}^{n}\right\rangle=\left\langle
g_{+}\right\rangle\left\langle{\mathfrak{p}}^{(n)}g_{-}{\circ}{\mathscr{S}}_{2}^{n}\right\rangle+\left\langle\tilde{g}_{+}{\mathfrak{p}}^{(n)}g_{-}{\circ}{\mathscr{S}}_{2}^{n}\right\rangle,$

where the second term can be bounded using Lemma C.1. If we write the first
term using (C.3), we see we need to estimate
$\langle\Delta_{i}^{(n-i)}\rangle$. We can write, for any $k\geq 1$,

$\displaystyle\Delta_{i}^{(k)}(\theta,\psi)=({\mathfrak{p}}_{i}(\theta,\psi)-\langle{\mathfrak{p}}_{i}\rangle(\theta))$
(C.8)
$\displaystyle\times\Bigl{(}{\mathfrak{p}}^{(k-1)}_{i+1}({\mathscr{S}}_{2}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k}(\theta,\psi))-{\mathfrak{p}}^{(k-1)}_{i+1}(\overline{\mathscr{S}}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{\mathscr{S}}(\theta,\psi)))\Bigr{)}$
$\displaystyle+({\mathfrak{p}}_{i}(\theta,\psi)-\langle{\mathfrak{p}}_{i}\rangle(\theta))\,{\mathfrak{p}}^{(k-1)}_{i+1}(\overline{\mathscr{S}}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{\mathscr{S}}(\theta,\psi)))$
$\displaystyle+\langle{\mathfrak{p}}_{i}\rangle(\theta)\Bigl{(}{\mathfrak{p}}^{(k-1)}_{i+1}({\mathscr{S}}_{2}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k}(\theta,\psi))-{\mathfrak{p}}^{(k-1)}_{i+1}(\overline{\mathscr{S}}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{\mathscr{S}}(\theta,\psi)))\Bigr{)}.$

We have
$\|({\mathfrak{p}}_{i}(\theta,\psi)-\langle{\mathfrak{p}}_{i}\rangle(\theta))\|_{\infty}\leq
C\rho$, while, reasoning like when studying (C.6) and (C.7), we get

$\|{\mathfrak{p}}^{(k-1)}_{i+1}({\mathscr{S}}_{2}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k}(\theta,\psi))-{\mathfrak{p}}^{(k-1)}_{i+1}(\overline{\mathscr{S}}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{\mathscr{S}}(\theta,\psi)))\|_{\infty}\leq
C\rho(1-\rho\,\gamma^{\prime})^{k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,1},$

so that the average of the first contribution in the r.h.s. of (C.8) is
bounded by
$C\rho^{2}(1-\rho\,\gamma^{\prime})^{k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,1}$.
A similar bound for the contribution in the third line of (C.8) is obtained as
in the proof of Lemma C.1, by using (C.6) and (C.7), with the gain of a
further factor $\rho$ because of the factor
${\mathfrak{p}}_{i}-\langle{\mathfrak{p}}_{i}\rangle$. As to the contribution
in the last line, we use (C.3) twice and write

$\displaystyle{\mathfrak{p}}^{(k-1)}_{i+1}({\mathscr{S}}_{2}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k}(\theta,\psi))-{\mathfrak{p}}^{(k-1)}_{i+1}(\overline{\mathscr{S}}(\theta,\psi))\,g_{-}({\mathscr{S}}_{2}^{k-1}(\overline{\mathscr{S}}(\theta,\psi)))$
(C.9)
$\displaystyle\quad=\sum_{j=0}^{k-2}\Bigl{(}\langle{\mathfrak{p}}\rangle^{(j)}_{i+1}(G_{2}(\theta,\psi))\Delta^{(k-1-j)}_{j}(\overline{\mathscr{S}}^{j}({\mathscr{S}}_{2}(\theta,\psi)))-\langle{\mathfrak{p}}\rangle^{(j)}_{i+1}(\overline{G}(\theta))\Delta^{(k-1-j)}_{j}(\overline{\mathscr{S}}^{j+1}(\theta,\psi))\Bigr{)}$
$\displaystyle\quad+\langle{\mathfrak{p}}\rangle^{(k-1)}_{i+1}(G_{2}(\theta,\psi))\,g_{-}(\overline{\mathscr{S}}^{k-1}({\mathscr{S}}_{2}(\theta,\psi)))-\langle{\mathfrak{p}}\rangle^{(k-1)}_{i+1}(\overline{G}(\theta))\,g_{-}(\overline{\mathscr{S}}^{k}(\theta,\psi)).$

Thus, we apply Lemma C.3 first to the last line, which gives

$\displaystyle\left|\left\langle\langle{\mathfrak{p}}\rangle^{(k-1)}_{i+1}{\circ}G_{2}\,g_{-}{\circ}\overline{\mathscr{S}}^{k-1}{\circ}{\mathscr{S}}_{2}-\langle{\mathfrak{p}}\rangle^{(k-1)}_{i+1}{\circ}\overline{G}\,g_{-}{\circ}\overline{\mathscr{S}}^{k}\right\rangle\right|$
$\displaystyle\qquad\qquad\qquad\leq
C\rho(1-\rho\,\gamma^{\prime})^{k}\bigl{(}(1+\alpha_{0}k)\lambda^{-\alpha_{0}k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}^{-}+\rho|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g_{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}\bigr{)}\,,$

and then to the second line, with $g_{-}=\Delta^{(k-1-j)}_{j}$, so as to
obtain

$\displaystyle\left\langle\langle{\mathfrak{p}}\rangle^{(j)}_{i+1}{\circ}G_{2}\,\Delta^{(k-1-j)}_{j}{\circ}\overline{\mathscr{S}}^{j}{\circ}{\mathscr{S}}_{2}-\langle{\mathfrak{p}}\rangle^{(j)}_{i+1}{\circ}\overline{G}\,\Delta^{(k-1-j)}_{j}{\circ}\overline{\mathscr{S}}^{j+1}\right\rangle$
$\displaystyle\qquad\qquad\qquad\leq
C\rho(1-\rho\,\gamma^{\prime})^{j}\left((1+\alpha_{0}j)\lambda^{-\alpha_{0}j}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\Delta^{(k-1-j)}_{j}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}^{-}+\rho|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\Delta^{(k-1-j)}_{j}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}^{-}\right)\,,$

where
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\Delta^{(k-1-j)}_{j}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}^{-}$
and hence also
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\Delta^{(k-1-j)}_{j}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}^{-}$
are bounded as discussed in Remark C.2. This concludes the proof of
Proposition 7.30.

##  Appendix D Proof of some technical results

Recall that, as mentioned at the beginning of Appendix C.2, ${\mathscr{S}}$,
${\mathscr{S}}_{1}$ and ${\mathscr{S}}_{2}$ are a shortened notation for the
corresponding extended maps.

###  D.1 Derivative of the auxiliary map: proof of Proposition 7.33

In the light of Remark 7.26, we may and so assume that $\rho<\rho_{0}$. First
of all we note that

$\displaystyle\partial_{\theta}\,(p_{2}^{(n)}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n})$
$\displaystyle=\partial_{\theta}p_{2}^{(n)}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n}+p_{2}^{(n)}\,\partial_{\theta}(q_{2}{\circ}{\mathscr{S}}_{2}^{n}),$
(D.1a)
$\displaystyle\partial_{\theta}\,(\overline{p}^{(n)}\,\overline{q}{\circ}\overline{G}^{n})$
$\displaystyle=\partial_{\theta}\overline{p}^{(n)}\,\overline{q}{\circ}\overline{G}^{n}+\overline{p}^{(n)}\,\partial_{\theta}(\overline{q}{\circ}\overline{G}^{n}),$
(D.1b)

where both $\partial_{\theta}p_{2}^{(n)}$ and
$\partial_{\theta}\overline{p}^{(n)}$ can be written according to (B.1a), so
that we can bound separately

$\langle\partial_{\theta}p_{2}^{(n)}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n}\rangle-\partial_{\theta}\overline{p}^{(n)}\,\overline{q}{\circ}\overline{G}^{n},\qquad\langle
p_{2}^{(n)}\,\partial_{\theta}(q_{2}{\circ}{\mathscr{S}}_{2}^{n})\rangle-\overline{p}^{(n)}\,\partial_{\theta}(\overline{q}{\circ}\overline{G}^{n}).$
(D.2)

Since
$\partial_{\theta}({\mathscr{S}}_{2})^{n}_{\theta}=(\partial_{\theta}G_{2})^{(n)}$
and $\partial_{\theta}\overline{G}^{n}=(\partial_{\theta}\overline{G})^{(n)}$,
we bound the first contribution in (D.2) as

$\displaystyle\bigl{|}\langle\partial_{\theta}p_{2}^{(n)}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n}\rangle-\partial_{\theta}\overline{p}^{(n)}\,\overline{q}{\circ}\overline{G}^{n}\bigr{|}$
$\displaystyle\leq\sum_{k=0}^{n-1}\Bigl{|}\langle(p_{2}\,\partial_{\theta}G_{2})^{(k)}\bigl{(}\partial_{\theta}p_{2}\,p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n-k}\bigr{)}{\circ}{\mathscr{S}}_{2}^{k}\rangle$
(D.3) $\displaystyle\qquad\qquad-\langle
p_{2}\,\partial_{\theta}G_{2}\rangle^{(k)}\langle\partial_{\theta}p_{2}\,p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n-k}\rangle{\circ}\overline{G}^{k}\Bigr{|}$
$\displaystyle+\sum_{k=0}^{n-1}\bigl{|}\langle
p_{2}\,\partial_{\theta}G_{2}\rangle^{(k)}\bigr{|}\bigl{|}\bigl{(}\langle\partial_{\theta}p_{2}\,p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n-k}\rangle{\circ}\overline{G}^{k}$
$\displaystyle\qquad\qquad-\langle\partial_{\theta}p_{2}\rangle\langle
p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\rangle\langle
q_{2}\rangle{\circ}\overline{G}^{n-k}\bigr{)}{\circ}\overline{G}^{k}\bigr{|}$
$\displaystyle+\sum_{k=0}^{n-1}\bigl{|}\langle
p_{2}\partial_{\theta}G_{2}\rangle^{(k)}\bigl{(}\langle\partial_{\theta}p_{2}\rangle\langle
p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\rangle\langle
q_{2}\rangle{\circ}\overline{G}^{n-k}\bigr{)}{\circ}\overline{G}^{k}$
$\displaystyle\qquad\qquad-(\overline{p}\,\partial_{\theta}\overline{G})^{(k)}\bigl{(}\partial_{\theta}\overline{p}\,\overline{p}^{(n-1-k)}{\circ}\overline{G}\,\overline{q}{\circ}\overline{G}^{n-k}\bigr{)}{\circ}\overline{G}^{k}\big{|}.$

Next, we use Proposition 7.30, with $k$ instead of $n$, $g_{+}=1$,
${\mathfrak{p}}_{i}=p_{2}\partial_{\theta}G_{2}$ for $i=1,\ldots,k$ and
$g_{-}=\partial_{\theta}p_{2}\,p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\,q_{2}{\circ}{\mathscr{S}}^{n-k}$,
to bound the contributions in the first two lines of (D.3) with

$\sum_{k=1}^{n}C(1-\rho\,\gamma^{\prime})^{k}\bigl{(}\rho+k\rho^{2}\bigr{)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}p_{2}\,p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n-k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-},$
(D.4)

and with $n-1-k$ instead of $n$, $g_{+}=\partial_{\theta}p_{2}$,
${\mathfrak{p}}_{i}=p_{2}{\circ}{\mathscr{S}}_{2}$ for $i=1,\ldots,n-k-1$ and
$g_{-}=q_{2}$, to bound the contributions in the third and fourth line of
(D.3) with

$\sum_{k=0}^{n-1}C(1-\rho\,\gamma^{\prime})^{k}\Bigl{(}(n-1-k)\lambda^{-\alpha_{0}(n-1-k)}+\bigl{(}\rho+(n-1-k)\rho^{2}\bigr{)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}p_{2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0}}^{+}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|q_{2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}\Bigr{)},$
(D.5)

and, after writing

$\displaystyle\bigl{|}\langle(p_{2}\,\partial_{\theta}G_{2})^{(k)}\rangle\bigl{(}\langle\partial_{\theta}p_{2}\rangle\langle
p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\rangle\langle
q_{2}\rangle{\circ}\overline{G}^{n-k}\bigr{)}{\circ}\overline{G}^{k}-(\overline{p}\,\partial_{\theta}\overline{G})^{(k)}\bigl{(}\partial_{\theta}\overline{p}\,\,\overline{p}^{(n-1-k)}{\circ}\overline{G}\,\overline{q}{\circ}\overline{G}^{n-k}\bigr{)}{\circ}\overline{G}^{k}\bigl{|}$
$\displaystyle\qquad\leq\bigl{|}\langle
p_{2}\,\partial_{\theta}G_{2}\rangle^{(k)}-(\overline{p}\,\partial_{\theta}\overline{G})^{(k)}\bigr{|}\bigl{|}\bigl{(}\langle\partial_{\theta}p_{2}\rangle\langle
p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\rangle\langle
q_{2}\rangle{\circ}\overline{G}^{n-k}\bigr{)}{\circ}\overline{G}^{k}\bigr{|}$
$\displaystyle\qquad\qquad+\bigl{|}(\overline{p}\,\partial_{\theta}\overline{G})^{(k)}\bigr{|}\bigl{|}(\langle\partial_{\theta}p_{2}\rangle-\partial_{\theta}\overline{p}\bigr{)}{\circ}\overline{G}^{k}\bigr{|}\bigl{|}(\langle
p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\rangle\langle
q_{2}\rangle{\circ}\overline{G}^{n-k}\bigr{)}{\circ}\overline{G}^{k}\bigr{|}\phantom{\Bigr{(}}$
$\displaystyle\qquad\qquad+\bigl{|}(\overline{p}\,\partial_{\theta}\overline{G})^{(k)}\,\partial_{\theta}\overline{p}{\circ}\overline{G}^{k}\bigr{|}\bigl{|}\bigl{(}\langle
p_{2}^{(n-1-k)}{\circ}{\mathscr{S}}_{2}\rangle-\overline{p}^{(n-1-k)}{\circ}\overline{G}\bigr{)}{\circ}\overline{G}^{k}\bigr{|}\bigr{|}\langle
q_{2}\rangle{\circ}\overline{G}^{n}\bigr{|}\phantom{\Bigr{(}}$
$\displaystyle\qquad\qquad+\bigl{|}(\overline{p}\,\partial_{\theta}\overline{G})^{(k)}\,\bigl{(}\partial_{\theta}\overline{p}\,\,\overline{p}^{(n-1-k)}{\circ}\overline{G}\bigr{)}{\circ}\overline{G}^{k}\bigr{|}\bigl{|}\langle
q_{2}\rangle{\circ}\overline{G}^{n}-\overline{q}{\circ}\overline{G}^{n}\bigl{|},\phantom{\Bigr{(}}$

we rely once more Proposition 7.30 to bound the last two lines of (D.3) with

$\displaystyle\sum_{k=0}^{n-1}C(1\\!-\\!\rho\gamma^{\prime})^{k}\Bigl{(}{(\rho+k\rho^{2})}\|\partial_{\theta}p_{2}\|_{\infty}\|p_{2}^{(n-1-k)}\|_{\infty}\|q_{2}\|_{\infty}+\rho\,\|\langle\partial_{\theta}p_{2}\rangle\\!-\\!\partial_{\theta}\overline{p}\|_{\infty}\|p_{2}^{(n-1-k)}\|_{\infty}\|q_{2}\|_{\infty}$
(D.6)
$\displaystyle\qquad\qquad+{(\rho+(n-k-1)\,\rho^{2})}\,(1-\rho\,\gamma^{\prime})^{n-k-1}\|q_{2}\|_{\infty}+\rho\,(1-\rho\,\gamma^{\prime})^{n-1-k}\|\langle
q_{2}\rangle-\overline{q}\|_{\infty}\Bigr{)}.$

Therefore, thanks to (7.57) and (7.60), which yield

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}p_{2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}\leq
C\rho,\qquad|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\langle
p_{2}\rangle-\overline{p}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,1}\leq
C\rho^{2},\qquad\|q_{2}\|_{\infty}\leq C\rho,\qquad\|\langle
q_{2}\rangle-\overline{q}\|_{\infty}\leq C\rho^{2},$

by collecting together the bounds (D.4) to (D.6) and using (2.10), we obtain

$\bigl{|}\langle\partial_{\theta}p_{2}^{(n)}\,q_{2}{\circ}{\mathscr{S}}_{2}^{n}\rangle-\partial_{\theta}\overline{p}^{(n)}\,\overline{q}{\circ}\overline{G}^{n}\bigr{|}\leq\sum_{k=0}^{n-1}C(1-\rho\,\gamma^{\prime})^{n}\rho^{3}\left(1+k\rho+k^{2}\rho^{2}\right).$
(D.7)

Next, we consider the second contribution in (D.2), that we rewrite as

$\displaystyle\langle
p_{2}^{(n)}\,\partial_{\theta}(q_{2}{\circ}{\mathscr{S}}_{2}^{n})\rangle-\overline{p}^{(n)}\,\partial_{\theta}(\overline{q}{\circ}\overline{G}^{n})$
$\displaystyle\qquad=\Bigl{(}\langle
p_{2}^{(n)}\,\partial_{\theta}(q_{2}{\circ}{\mathscr{S}}_{2}^{n})\rangle-\langle
p_{2}\rangle^{(n)}\langle\partial_{\theta}(q_{2}{\circ}\overline{G}^{n})\rangle\Bigr{)}+\Bigl{(}\langle
p_{2}\rangle^{(n)}\langle\partial_{\theta}(q_{2}{\circ}\overline{G}^{n}\rangle-\overline{p}^{(n)}\,\partial_{\theta}(\overline{q}{\circ}\overline{G}^{n})\Bigr{)},$

and, using that

$\displaystyle\partial_{\theta}(q_{2}{\circ}{\mathscr{S}}_{2}^{n})=(\partial_{\theta}({\mathscr{S}}_{2})_{\theta})^{(n)}(\partial_{\theta}q_{2}){\circ}{\mathscr{S}}_{2}^{n},$
$\displaystyle\partial_{\theta}(q_{2}{\circ}\overline{G}^{n})=(\partial_{\theta}\overline{G})^{(n)}\partial_{\theta}q_{2}{\circ}\overline{G}^{n},$
$\displaystyle\partial_{\theta}(\overline{q}{\circ}\overline{G}^{n})=(\partial_{\theta}\overline{G})^{(n)}(\partial_{\theta}\overline{q}){\circ}\overline{G}^{n},$

we apply first Proposition 7.30, with
${\mathfrak{p}}_{i}=p_{2}\,\partial_{\theta}{\mathscr{S}}_{2}$, $g_{+}=1$ and
$g_{-}=q_{2}$ to bound

$\bigl{|}\langle(p_{2}\,\partial_{\theta}({\mathscr{S}}_{2})_{\theta})^{(n)}(\partial_{\theta}q_{2}){\circ}{\mathscr{S}}_{2}^{n}\rangle-\langle
p_{2}\,\partial_{\theta}({\mathscr{S}}_{2})_{\theta}\rangle^{(n)}\langle\partial_{\theta}q_{2}\rangle{\circ}\overline{G}^{n}\bigr{|}\leq
C(1-\rho\,\gamma^{\prime})^{n}{\left(\rho+\rho^{2}n\right)}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}q_{2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3},$

then reason as in (7.55) to get

$\displaystyle\left|\langle
p_{2}\partial_{\theta}({\mathscr{S}}_{2})_{\theta}\rangle^{(n)}\langle\partial_{\theta}q_{2}\rangle{\circ}\overline{G}^{n}-(\overline{p}\,\partial_{\theta}\overline{G})^{(n)}\,(\partial_{\theta}\overline{q}){\circ}\overline{G}^{n}\right|$
$\displaystyle\qquad\leq\langle\partial_{\theta}q_{2}\rangle{\circ}\overline{G}^{n}\sum_{i=0}^{n-1}\langle
p_{2}\,\partial_{\theta}({\mathscr{S}}_{2})_{\theta}\rangle^{(i)}\left(\langle
p_{2}\partial_{\theta}({\mathscr{S}}_{2})_{\theta}\rangle-\overline{p}\,\partial_{\theta}\overline{G}\right){\circ}\overline{G}^{i}\,(\overline{p}\,\partial_{\theta}\overline{G})^{(n-i-2)}{\circ}\overline{G}^{i+1}$
$\displaystyle\qquad\qquad+(\overline{p}\,\partial_{\theta}\overline{G})^{(n)}\left(\langle\partial_{\theta}q_{2}\rangle{\circ}\overline{G}^{n}-\partial_{\theta}\overline{q}{\circ}\overline{G}^{n}\right)$
$\displaystyle\qquad\leq
C(1-\rho\,\gamma^{\prime})^{n}n\,\|p_{2}\partial_{\theta}({\mathscr{S}}_{2})_{\theta}-\overline{p}\,\partial_{\theta}\overline{G}\|_{\infty}\,\|\partial
q_{2}\|_{\infty}+C(1-\rho\,\gamma^{\prime})^{n}\,\|\langle\partial_{\theta}q_{2}\rangle-\partial_{\theta}\bar{q}\|_{\infty}\phantom{\sum_{i=0}^{n-1}}$
$\displaystyle\qquad\leq C(1-\rho\,\gamma^{\prime})^{n}n\rho^{2},$

so that eventually we obtain

$\bigl{|}\langle
p_{2}^{(n)}\,\partial_{\theta}(q_{2}{\circ}{\mathscr{S}}_{2}^{n})-\overline{p}^{(n)}\,\partial_{\theta}(\overline{q}{\circ}\overline{G}^{n})\rangle\bigr{|}\leq
C\rho.$ (D.8)

and hence, summing together (D.7) and (D.8), the first bound in (7.63)
follows.

The second bound in (7.63) is obtained in a similar way by reasoning as in the
second part of the proof of Proposition 7.32 and using (D.7) and (D.8) instead
of (7.59).

###  D.2 Comparing the translated and auxiliary maps I: proof of Lemma 7.37

As in Appendix D.1, we consider explicitly only the case $\rho<\rho_{0}$. Note
that all the proofs from here on, until the end of this appendix, undergo
major simplifications if ${\mathscr{S}}$ is restricted to the set
$\Lambda_{2r_{2}}$ as discussed in Remark 7.40; in particular all term which
involve the function $c_{0}$ vanish identically.

We follow the strategy we have outlined in Remark 7.36, by rearranging sums
where differences $W{\circ}A_{0}^{i+1}-W{\circ}A_{0}^{i}$ appear (see the
equations (D.10) to (D.14) below) in such a way that the new summands contain
differences of more regular functions (see Lemmas D.1 and D.4 below).

Thus, if we define, for $k\geq 0$,

$\displaystyle D_{{\mathfrak{p}},1,k}(\theta,\psi)$
$\displaystyle:=\int_{0}^{1}{\rm
d}t\,\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{1}^{k})\,(G_{2}(\theta,\psi)+t\,\rho\,\zeta(\theta,\psi),A_{0}\psi),$
(D.9a) $\displaystyle D_{{\mathfrak{p}},2,k}(\theta,\psi)$
$\displaystyle:=\int_{0}^{1}{\rm
d}t\,(1-t)\,\partial^{2}_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{1}^{k})\,(G_{2}(\theta,\psi)+t\,\rho\,\zeta(\theta,\psi),A_{0}\psi).$
(D.9b) $\displaystyle R_{{\mathfrak{p}},1,k}(\theta,\psi)$
$\displaystyle:=\sum_{i=0}^{k}\left(\partial_{\theta}\bigl{(}(D_{{\mathfrak{p}},1,k-i}\,\zeta){\circ}{\mathscr{S}}_{2}^{i}\bigr{)}\right)(\theta,\psi),$
(D.9c) $\displaystyle R_{{\mathfrak{p}},2,k}(\theta,\psi)$
$\displaystyle:=\sum_{i=0}^{k}\left(\left(D_{{\mathfrak{p}},2,k-i}\,\zeta^{2}\right){\circ}{\mathscr{S}}_{2}^{i}\right)(\theta,\psi),$
(D.9d)

we can write, for $n\geq 1$,

$\displaystyle{\mathfrak{p}}{\circ}{\mathscr{S}}_{1}^{n}-{\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{n}$
$\displaystyle=\sum_{i=0}^{n-1}\left({\mathfrak{p}}{\circ}{\mathscr{S}}_{1}^{n-i}{\circ}{\mathscr{S}}_{2}^{i}-{\mathfrak{p}}{\circ}{\mathscr{S}}_{1}^{n-1-i}{\circ}{\mathscr{S}}_{2}^{i+1}\right)$
(D.10)
$\displaystyle=\rho\sum_{i=0}^{n-1}\bigl{(}D_{{\mathfrak{p}},1,n-1-i}\,\zeta\bigr{)}{\circ}{\mathscr{S}}_{2}^{i}$
$\displaystyle=\rho\sum_{i=0}^{n-1}\bigl{(}\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{1}^{n-1-i}){\circ}{\mathscr{S}}_{2}\,\zeta\bigr{)}{\circ}{\mathscr{S}}_{2}^{i}+\rho^{2}R_{{\mathfrak{p}},2,n-1},$

and, using the second line of (D.10), with $n-i-1$ instead of $n$, to write

$\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{1}^{n-1-i})=\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{n-1-i})+\rho\sum_{i=0}^{n-1-i}\partial_{\theta}\bigl{(}\bigl{(}D_{{\mathfrak{p}},1,n-i-1-j}\,\zeta\bigr{)}{\circ}{\mathscr{S}}_{2}^{j}\bigr{)}$

in the last line, we obtain

${\mathfrak{p}}{\circ}{\mathscr{S}}_{1}^{n}-{\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{n}=\rho\\!\sum_{i=0}^{n-1}\bigl{(}\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{n-1-i}){\circ}{\mathscr{S}}_{2}\,\zeta\bigr{)}{\circ}{\mathscr{S}}_{2}^{i}+\rho^{2}\sum_{i=0}^{n-1}\left(R_{{\mathfrak{p}},1,n-1-i}{\circ}{\mathscr{S}}_{2}\,\zeta\right){\circ}{\mathscr{S}}_{2}^{i}+\rho^{2}R_{{\mathfrak{p}},2,n-1}.$
(D.11)

Note that, by Lemma 6.9 and Remark 6.13, for all $n\geq 1$ one has

$\|R_{{\mathfrak{p}},2,n-1}\|_{\infty}\leq
C,\qquad\left\langle|R_{{\mathfrak{p}},2,n-1}|\right\rangle\leq C.$ (D.12)

Next, using (7.68) and recalling (7.46), if we set

$E_{{\mathfrak{p}},r,k}:=\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k}){\circ}{\mathscr{S}}_{2}\,c_{r},\qquad
r=0,1,2,$ (D.13)

then in the first sum on the r.h.s. of (D.11) we can write

$\displaystyle\left(\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{n-i-1}){\circ}{\mathscr{S}}_{2}\,\zeta\right){\circ}{\mathscr{S}}_{2}^{i}$
(D.14)
$\displaystyle\qquad=\rho^{-1}E_{{\mathfrak{p}},0,n-i-1}{\circ}{\mathscr{S}}_{2}^{i}\,(W{\circ}A_{0}^{i+1}-W{\circ}A_{0}^{i})+\sum_{r=1}^{2}E_{{\mathfrak{p}},r,n-i-1}{\circ}{\mathscr{S}}_{2}^{i}\,\left(W{\circ}A_{0}^{i}\right)^{r}.$

The following result is easily checked.

######  Lemma D.1.

Let $\\{E_{k}\\}_{k=0}^{n-1}$ be a set of functions
$E_{k}\\!:\Omega\to\Omega$, with $n\geq 1$. For $m\in\mathds{N}$ such that
$1\leq m\leq n$, define

$\Delta E_{m,0}=E_{0}{\circ}{\mathscr{S}}_{2}^{-1},\quad\Delta
E_{m,k}=E_{k}{\circ}{\mathscr{S}}_{2}^{-1}-E_{k-1},\,k=1,\ldots,m-1,\quad\Delta
E_{m,m}=-E_{m-1},$ (D.15)

where $\Delta E_{m,k}$ with $0<k<m$ are meaningful only if $m\geq 2$. Then,
for any $m$ as above, one has

$\sum_{i=0}^{m-1}E_{m-1-i}{\circ}{\mathscr{S}}_{2}^{i}\,(W{\circ}A_{0}^{i+1}-W{\circ}A_{0}^{i})\\\
=\sum_{i=0}^{m}\Delta
E_{m,m-i}{\circ}{\mathscr{S}}_{2}^{i}\,W{\circ}A_{0}^{i}.$ (D.16)

######  Remark D.2.

_If the functions $E_{k}$ in Lemma D.1 are such that_

$\displaystyle|{\hskip-0.85355pt}|{\hskip-0.85355pt}|E_{k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}\leq(1-\rho\,\gamma^{\prime})^{k}\,\Gamma_{0},\qquad
k=0,\ldots,n-1,$ (D.17a)
$\displaystyle|{\hskip-0.85355pt}|{\hskip-0.85355pt}|E_{k}{\circ}{\mathscr{S}}_{2}^{-1}-E_{k-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}\leq\rho\,(1-\rho\,\gamma^{\prime})^{k}\,\Gamma_{0},\qquad
k=1,\ldots,n-1,$ (D.17b)

_for some constant $\Gamma_{0}$, then one finds_

$\sum_{i=0}^{n}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\Delta
E_{n,n-i}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}\leq
C\,\Gamma_{0}.$ (D.18)

_Analogously, if_

$\displaystyle\|E_{k}\|_{\infty}\leq(1-\rho\,\gamma^{\prime})^{k}\,\Gamma_{0},\qquad
k=0,\ldots,n-1,$ (D.19a)
$\displaystyle\|E_{k}{\circ}{\mathscr{S}}_{2}^{-1}-E_{k-1}\|_{\infty}\leq\rho\,(1-\rho\,\gamma^{\prime})^{k}\,\Gamma_{0},\qquad
k=1,\ldots,n-1,$ (D.19b)

_one has_

$\sum_{i=0}^{n}\|\Delta E_{n,n-i}\|_{\infty}\leq C\,\Gamma_{0}.$ (D.20)

######  Remark D.3.

_The functions $E_{{\mathfrak{p}},0,k}$ introduced in (D.13) satisfy the
conditions (D.17) with
$\Gamma_{0}=C|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}$.
Indeed, using that
$\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k})=\partial_{\theta}{\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k}\partial_{\theta}({\mathscr{S}}_{2}^{k})_{\theta}$,
one has_

$\displaystyle|{\hskip-0.85355pt}|{\hskip-0.85355pt}|E_{{\mathfrak{p}},0,k}{\circ}{\mathscr{S}}_{2}^{-1}-E_{{\mathfrak{p}},0,k-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}$
$\displaystyle\qquad\leq|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k})|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|c_{0}{\circ}{\mathscr{S}}_{2}^{-1}-c_{0}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}+|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k}){\circ}{\mathscr{S}}_{2}^{-1}-\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k-1})|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|c_{0}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}$
$\displaystyle\qquad\leq|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}({\mathscr{S}}_{2}^{k})_{\theta}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|c_{0}{\circ}{\mathscr{S}}_{2}^{-1}-c_{0}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}$
$\displaystyle\qquad+|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}({\mathscr{S}}_{2}^{k-1})_{\theta}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}({\mathscr{S}}_{2})_{\theta}-\mathds{1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|c_{0}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}$
$\displaystyle\qquad\leq
C\rho\,(1-\rho\,\gamma^{\prime})^{k}\,|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3},$

_as a consequence of the bounds ( 7.54) in Lemma 7.27. _

Lemma D.1 is extended immediately as follows.

######  Lemma D.4.

Let $\\{E_{k}\\}_{k=0}^{n-1}$ and, for $1\leq m\leq n$, $\\{\Delta
E_{m,k}\\}_{k=1}^{m}$ be as in Lemma D.1, and let $\\{F_{k}\\}_{k=0}^{n}$ be a
set of functions $F_{k}\\!:\Omega\to\Omega$. Then, for any $m$ as above, one
has

$\displaystyle\sum_{i=0}^{m-1}E_{m-1-i}{\circ}{\mathscr{S}}_{2}^{i}\,F_{i}(W{\circ}A_{0}^{i+1}-W{\circ}A_{0}^{i})$
(D.21) $\displaystyle\qquad=\sum_{i=0}^{m}\Delta
E_{m,m-i}{\circ}{\mathscr{S}}_{2}^{i}\,F_{i}W{\circ}A_{0}^{i}+\sum_{i=1}^{m}E_{m-i}{\circ}{\mathscr{S}}_{2}^{i-1}\bigl{(}F_{i-1}-F_{i}\bigr{)}W{\circ}A_{0}^{i}.$

######  Remark D.5.

_It is easy to check that_

$\displaystyle\|\partial_{\theta}({\mathscr{S}}_{2}^{k})_{\theta}\,c_{r}{\circ}{\mathscr{S}}_{2}^{-1}-\partial_{\theta}({\mathscr{S}}_{2}^{k})_{\theta}{\circ}{\mathscr{S}}_{2}\,c_{r}\|_{\infty}$
$\displaystyle\qquad\leq\|\partial_{\theta}({\mathscr{S}}_{2}^{k})_{\theta}\|_{\infty}\|c_{r}{\circ}({\mathscr{S}}_{2}^{-1})_{\theta}-c_{r}\|_{\infty}+\|\partial_{\theta}^{2}({\mathscr{S}}_{2}^{k})_{\theta}\|_{\infty}\|({\mathscr{S}}_{2})_{\theta}-\mathds{1}\|_{\infty}\|\,c_{r}\|_{\infty}$
$\displaystyle\qquad\leq C\rho(1-\rho\,\gamma^{\prime})^{k},$

_and, analogously,_

$\displaystyle\|\partial_{\theta}\bigl{(}D_{{\mathfrak{p}},1,k}\,c_{r}\bigr{)}{\circ}{\mathscr{S}}_{2}^{-1}-\partial_{\theta}\bigl{(}D_{{\mathfrak{p}},1,k-1}\,c_{r}\bigr{)}\|_{\infty}$
$\displaystyle\qquad\leq\|\partial_{\theta}\bigl{(}D_{{\mathfrak{p}},1,k}\,c_{r}\bigr{)}\|_{\infty}\|{\circ}{\mathscr{S}}_{2}^{-1}-\mathds{1}\|_{\infty}+\|\partial_{\theta}\bigl{(}D_{{\mathfrak{p}},1,k}\,c_{r}\bigr{)}-\partial_{\theta}\bigl{(}D_{{\mathfrak{p}},1,k-1}\,c_{r}\bigr{)}\|_{\infty}$
$\displaystyle\qquad\leq
C\rho\,(1-\rho\,\gamma^{\prime})^{k+j}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}.$

_Therefore the functions $Q_{r,k}$ and $Q_{{\mathfrak{p}},r,k}$, for
$r=0,1,2$, satisfy the conditions (D.19), with $\Gamma_{0}=C$ and
$\Gamma_{0}=C|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}$
respectively, and hence, by Remark D.2, we can bound_

$\|\Delta Q_{r,m,k}\|_{\infty}\leq
C\rho(1-\rho\,\gamma^{\prime})^{k},\qquad\|\Delta
Q_{{\mathfrak{p}},r,m,k}\|_{\infty}\leq
C\rho(1-\rho\,\gamma^{\prime})^{k}\,|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3},$
(D.22)

_for $r=0,1$, $m\geq 2$ and $k=1,\ldots,m-1$. _

By Lemma D.1, after inserting (D.14) into (D.11), we obtain

$\sum_{i=0}^{n-1}E_{{\mathfrak{p}},0,n-1-i}{\circ}{\mathscr{S}}_{2}^{i}\,(W{\circ}A_{0}^{i+1}-W{\circ}A_{0}^{i})\\\
=\sum_{i=0}^{n}\Delta
E_{{\mathfrak{p}},0,n,n-i}{\circ}{\mathscr{S}}_{2}^{i}\,W{\circ}A_{0}^{i},$
(D.23)

where the functions $\Delta E_{{\mathfrak{p}},0,m,k}$ are defined as in (D.15)
with $E_{k}=E_{{\mathfrak{p}},0,k}$, so that,

$\sum_{i=0}^{n}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\Delta
E_{{\mathfrak{p}},0,n,n-i}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}^{-}\leq
C\,|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3},$
(D.24)

by Remarks D.2 and D.3. Furthermore, in the second sum on the r.h.s. of
(D.11), if we set

$Q_{{\mathfrak{p}},r,k}:=\partial_{\theta}\bigl{(}D_{{\mathfrak{p}},1,k}\,c_{r}\bigr{)},\qquad
Q_{r,k}:=\partial_{\theta}({\mathscr{S}}_{2}^{k})_{\theta}{\circ}{\mathscr{S}}_{2}\,c_{r},\qquad
r=0,1,2,$ (D.25)

and define $\Delta Q_{{\mathfrak{p}},r,m,k}$ and $\Delta Q_{r,m,k}$ according
to (D.15) with $E_{k}=Q_{{\mathfrak{p}},r,k}$ and $E_{k}=Q_{r,k}$,
respectively, we obtain

$\displaystyle\sum_{i=0}^{n-1}\left(R_{{\mathfrak{p}},1,n-1-i}{\circ}{\mathscr{S}}_{2}\,\zeta\right){\circ}{\mathscr{S}}_{2}^{i}=\sum_{i=0}^{n-1}\sum_{j=0}^{i}\left(\partial_{\theta}\left(\bigl{(}D_{{\mathfrak{p}},1,n-1-i}\,\zeta\bigr{)}{\circ}{\mathscr{S}}_{2}^{i-j}\right){\circ}{\mathscr{S}}_{2}\,\zeta\right){\circ}{\mathscr{S}}_{2}^{j}$
(D.26)
$\displaystyle\qquad=\sum_{i=0}^{n-1}\sum_{j=0}^{i}\sum_{r,s=1}^{2}Q_{{\mathfrak{p}},r,n-1-i}{\circ}{\mathscr{S}}_{2}^{i+1}Q_{s,i-j}{\circ}{\mathscr{S}}_{2}^{j}\,\bigl{(}W{\circ}A_{0}^{i+1}\bigr{)}^{r}\bigl{(}W{\circ}A_{0}^{j}\bigr{)}^{s}$
$\displaystyle\qquad+\rho^{-1}\sum_{i=0}^{n-1}\sum_{j=0}^{i+1}\sum_{r=1}^{2}Q_{{\mathfrak{p}},r,n-1-i}{\circ}{\mathscr{S}}_{2}^{i+1}\Delta
Q_{0,i+1,i+1-j}{\circ}{\mathscr{S}}_{2}^{j}\,\bigl{(}W{\circ}A_{0}^{i+1}\bigr{)}^{r}\,W{\circ}A_{0}^{j}$
$\displaystyle\qquad+\rho^{-1}\Biggl{(}\sum_{j=0}^{n-1}\sum_{i=0}^{n-j}\sum_{s=1}^{2}\Bigl{(}\Delta
Q_{{\mathfrak{p}},0,n-j,n-j-i}{\circ}{\mathscr{S}}_{2}^{i+1}Q_{s,i}\,W{\circ}A_{0}^{i+1}\Bigr{)}{\circ}{\mathscr{S}}_{2}^{j}\bigl{(}W{\circ}A_{0}^{j}\bigr{)}^{s}$
$\displaystyle\qquad\qquad+\sum_{j=0}^{n-1}\sum_{i=1}^{n-j}\sum_{s=1}^{2}\Bigl{(}Q_{{\mathfrak{p}},0,n-j-i}{\circ}{\mathscr{S}}_{2}^{i}\left(Q_{s,i-1}-Q_{s,i}\right)W{\circ}A_{0}^{i+1}\Bigr{)}{\circ}{\mathscr{S}}_{2}^{j}\bigl{(}W{\circ}A_{0}^{j}\bigr{)}^{s}\Biggr{)}$
$\displaystyle\qquad+\rho^{-2}\Biggl{(}\sum_{j=0}^{n-1}\sum_{i=0}^{n-j}\Bigl{(}\Delta
Q_{{\mathfrak{p}},0,n-j,n-j-i}{\circ}{\mathscr{S}}_{2}^{i+1}\Delta
Q_{0,i+1,i+1}W{\circ}A_{0}^{i+1}\Bigr{)}{\circ}{\mathscr{S}}_{2}^{j}\,\,W{\circ}A_{0}^{j}$
$\displaystyle\qquad\qquad+\sum_{j=0}^{n-1}\sum_{i=1}^{n-j}\Bigl{(}Q_{{\mathfrak{p}},0,n-j-i}{\circ}{\mathscr{S}}_{2}^{i}\bigl{(}\Delta
Q_{0,i,i}-\Delta
Q_{0,i+1,i+1}\bigr{)}\Bigr{)}{\circ}{\mathscr{S}}_{2}^{j}\,W{\circ}A_{0}^{i+j+1}\,W{\circ}A_{0}^{j}$
$\displaystyle\qquad\qquad+\sum_{i=0}^{n-1}Q_{{\mathfrak{p}},0,n-1-i}{\circ}{\mathscr{S}}_{2}^{i+1}\,\Delta
Q_{0,i+1,0}{\circ}{\mathscr{S}}_{2}^{i+1}\,\bigl{(}W{\circ}A_{0}^{i+2}-W{\circ}A_{0}^{i+1}\bigr{)}W{\circ}A_{0}^{i+1}\Biggr{)},$

where we have used Lemma D.1 to obtain the third line, Lemma D.4 to obtain the
fourth and fifth lines, and first Lemma D.1 and then Lemma D.4 to obtain the
last three lines.

######  Remark D.6.

_To bound the second to last line of ( D.26) we use that, for $m>k\geq 0$, we
have_

$\displaystyle\Delta Q_{0,m,k}-\Delta
Q_{0,m+1,k+1}=\left(Q_{0,k}{\circ}{\mathscr{S}}_{2}^{-1}-Q_{0,k-1}\right)-\left(Q_{0,k+1}{\circ}{\mathscr{S}}_{2}^{-1}-Q_{0,k}\right)$
$\displaystyle\qquad=\partial_{\theta}({\mathscr{S}}_{2}^{k-1})_{\theta}\left(\partial_{\theta}({\mathscr{S}}_{2})_{\theta}\,c_{0}{\circ}{\mathscr{S}}_{2}^{-1}-c_{0}\right)-\partial_{\theta}({\mathscr{S}}_{2}^{k})_{\theta}\left(\partial_{\theta}({\mathscr{S}}_{2})_{\theta}\,c_{0}{\circ}{\mathscr{S}}_{2}^{-1}-c_{0}\right)$
$\displaystyle\qquad=\left(\partial_{\theta}({\mathscr{S}}_{2}^{k-1})_{\theta}-\partial_{\theta}({\mathscr{S}}_{2}^{k})_{\theta}\right)\left(\partial_{\theta}({\mathscr{S}}_{2})_{\theta}\,c_{0}{\circ}{\mathscr{S}}_{2}^{-1}-c_{0}\right)$
$\displaystyle\qquad=\partial_{\theta}({\mathscr{S}}_{2}^{k-1})_{\theta}\left(\mathds{1}-\partial_{\theta}({\mathscr{S}}_{2})_{\theta}\right)\left(\left(\partial_{\theta}({\mathscr{S}}_{2})_{\theta}-\mathds{1}\right)c_{0}{\circ}{\mathscr{S}}_{2}^{-1}+\left(c_{0}{\circ}{\mathscr{S}}_{2}^{-1}-c_{0}\right)\right),$

_so that $\|\Delta Q_{0,m,k}-\Delta Q_{0,m,k+1}\|_{\infty}\leq
C\rho^{2}(1-\rho\,\gamma^{\prime})^{k}$. _

The bounds (7.54) in Lemma 7.27 and the second bound of (2.38) in Theorem 4,
together with the bounds in Remarks D.5 and D.6, yield that

$\displaystyle\left\|\sum_{i=0}^{n-1}\left(R_{{\mathfrak{p}},1,n-1-i}{\circ}{\mathscr{S}}_{2}\,\zeta\right){\circ}{\mathscr{S}}_{2}^{i}\right\|_{\infty}\leq
C\,\rho^{-2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3},$
(D.27a)
$\displaystyle\left\langle\left|\sum_{i=0}^{n-1}\left(R_{{\mathfrak{p}},1,n-1-i}{\circ}{\mathscr{S}}_{2}\,\zeta\right){\circ}{\mathscr{S}}_{2}^{i}\right|\right\rangle\leq
C\rho^{-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}{\mathfrak{p}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},3}.$
(D.27b)

In conclusion, we obtain (7.69) with
$C_{{\mathfrak{p}},n,k}=E_{{\mathfrak{p}},1,k-1}+\rho^{-1}\Delta
E_{{\mathfrak{p}},0,n,k}$ for $k=1,\ldots,n$ and
$C_{{\mathfrak{p}},n,0}=\Delta E_{{\mathfrak{p}},1,n,0}$, i.e.

$\displaystyle C_{{\mathfrak{p}},n,0}$
$\displaystyle:=\rho^{-1}\partial_{\theta}{\mathfrak{p}}\,c_{0}{\circ}{\mathscr{S}}_{2}^{-1},$
(D.28) $\displaystyle C_{{\mathfrak{p}},n,k}$
$\displaystyle:=\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k-1}){\circ}{\mathscr{S}}_{2}\,c_{1}$
$\displaystyle+\rho^{-1}\bigl{(}\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k})\,c_{0}{\circ}{\mathscr{S}}_{2}^{-1}-\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{k-1}){\circ}{\mathscr{S}}_{2}\,c_{0}\bigr{)},\quad
k=1,\ldots,n\\!-\\!1,$ $\displaystyle C_{{\mathfrak{p}},n,n}$
$\displaystyle:=\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{n-1}){\circ}{\mathscr{S}}_{2}\,{\mathfrak{c}}_{1}-\rho^{-1}\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{n-1})\,c_{0}{\circ}{\mathscr{S}}_{2}^{-1}$

so that the bound (7.70b) follows from the estimates (7.54) and (D.24), while

$\displaystyle\\!\\!\\!R_{{\mathfrak{p}},n}:=\sum_{i=0}^{n-1}\Bigl{(}\partial_{\theta}({\mathfrak{p}}{\circ}{\mathscr{S}}_{2}^{n-1-i}){\circ}{\mathscr{S}}_{2}\,c_{2}\Bigr{)}{\circ}{\mathscr{S}}_{2}^{i}\,W^{2}{\circ}A_{0}^{i}+\rho\sum_{i=0}^{n-1}\left(R_{{\mathfrak{p}},1,n-1-i}{\circ}{\mathscr{S}}_{2}\,\zeta\right){\circ}{\mathscr{S}}_{2}^{i}+\rho\,R_{{\mathfrak{p}},2,n-1}$
(D.29)

satisfies the bounds in the statement because of the bounds (D.12) and (D.27).

######  Remark D.7.

_While the functions $C_{{\mathfrak{p}},n,0},\ldots,C_{{\mathfrak{p}},n,n}$
depend on ${\mathscr{S}}_{2}$ alone, the function $R_{{\mathfrak{p}},n}$
involves both ${\mathscr{S}}_{1}$ and ${\mathscr{S}}_{2}$ through the
contributions $R_{{\mathfrak{p}},2,n-1}$ and $R_{{\mathfrak{p}},1,k}$. _

###  D.3 Comparing the translated and auxiliary maps II: proof of Proposition
7.50

Once more, we can confine ourselves to the case $\rho<\rho_{0}$. We have

$\partial_{\theta}\bigl{(}p_{1}^{(n)}q_{1}{\circ}{\mathscr{S}}_{1}^{n}\bigr{)}=\partial_{\theta}p_{1}^{(n)}\,q_{1}{\circ}{\mathscr{S}}_{1}^{n}+p_{1}^{(n)}\,\partial_{\theta}(q_{1}{\circ}{\mathscr{S}}_{1}^{n}),$
(D.30)

which, together with (D.1a), allows us to write the first contribution in
(7.52b) as

$\displaystyle\partial_{\theta}h_{1}-\partial_{\theta}h_{2}$
$\displaystyle=\sum_{n=0}^{\infty}\partial_{\theta}\,(p_{1}^{(n)}q_{1}{\circ}{\mathscr{S}}_{1}^{n})-\partial_{\theta}\,(p_{2}^{(n)}q_{2}{\circ}{\mathscr{S}}_{2}^{n})$
(D.31)
$\displaystyle=\sum_{n=0}^{\infty}(\partial_{\theta}p_{1}^{(n)}-\partial_{\theta}p_{2}^{(n)})\,(q_{1}{\circ}{\mathscr{S}}_{1}^{n}-q_{2}{\circ}{\mathscr{S}}_{2}^{n})+\sum_{n=0}^{\infty}(\partial_{\theta}p_{1}^{(n)}-\partial_{\theta}p_{2}^{(n)})\,q_{2}{\circ}{\mathscr{S}}_{2}^{n}$
$\displaystyle+\sum_{n=0}^{\infty}\partial_{\theta}p_{2}^{(n)}\,(q_{1}{\circ}{\mathscr{S}}_{1}^{n}-q_{2}{\circ}{\mathscr{S}}_{2}^{n})+\sum_{n=0}^{\infty}(p_{1}^{(n)}-p_{2}^{(n)})\,(\partial_{\theta}q_{1}{\circ}{\mathscr{S}}_{1}^{n}-\partial_{\theta}q_{2}{\circ}{\mathscr{S}}_{2}^{n})$
$\displaystyle+\sum_{n=0}^{\infty}(p_{1}^{(n)}-p_{2}^{(n)})\,\partial_{\theta}q_{2}{\circ}{\mathscr{S}}_{2}^{n}+\sum_{n=0}^{\infty}p_{2}^{(n)}\,(\partial_{\theta}q_{1}{\circ}{\mathscr{S}}_{1}^{n}-\partial_{\theta}q_{2}{\circ}{\mathscr{S}}_{2}^{n}).$

In (D.31) we can expand

$p_{1}^{(n)}-p_{2}^{(n)}=\sum_{k=0}^{n-1}p_{1}^{(k)}\bigl{(}p_{1}{\circ}{\mathscr{S}}_{1}^{k}-p_{2}{\circ}{\mathscr{S}}_{2}^{k}\bigr{)}\,p_{2}^{(n-k-1)}{\circ}{\mathscr{S}}_{2}^{k+1},$
(D.32)

according to (7.65a), and

$\displaystyle\partial_{\theta}p_{1}^{(n)}-\partial_{\theta}p_{2}^{(n)}$
$\displaystyle=\sum_{k=0}^{n-1}\bigl{(}\pi_{1}^{(k)}-\pi_{2}^{(k)}\bigr{)}\,\partial_{\theta}p_{2}{\circ}{\mathscr{S}}_{2}^{k}\,p_{2}^{(n-k-1)}{\circ}{\mathscr{S}}_{2}^{k+1}$
(D.33)
$\displaystyle+\sum_{k=0}^{n-1}\pi_{1}^{(k)}\bigl{(}\partial_{\theta}p_{1}{\circ}{\mathscr{S}}_{1}^{k}-\partial_{\theta}p_{2}{\circ}{\mathscr{S}}_{2}^{k}\bigr{)}p_{2}^{(n-k-1)}{\circ}{\mathscr{S}}_{2}^{k+1}$
$\displaystyle+\sum_{k=0}^{n-1}\pi_{1}^{(k)}\,\partial_{\theta}p_{1}{\circ}{\mathscr{S}}_{1}^{k}\bigl{(}p_{1}^{(n-k-1)}{\circ}{\mathscr{S}}_{1}^{k+1}-p_{2}^{(n-k-1)}{\circ}{\mathscr{S}}_{2}^{k+1}\bigr{)}$

with $\pi_{1}:=p_{1}\partial_{\theta}G_{1}$ and
$\pi_{2}:=p_{2}\partial_{\theta}G_{2}$, first taking into account (B.1a) and
then proceeding as in deriving (7.65a). Next, we apply (7.87b), (7.66b) and,
once more, (7.65a) to write in the first line of (D.33)

$\displaystyle\pi_{1}^{(k)}-\pi_{2}^{(k)}$
$\displaystyle=\sum_{i=0}^{k-1}\pi_{1}^{(i)}\bigl{(}\Delta_{\pi_{1},\pi_{2},W,i}+\rho\,{\mathfrak{a}}_{3}{\circ}{\mathscr{S}}_{1}^{i}\,{\mathfrak{f}}{\circ}A_{0}^{i}\bigr{)}\,\pi_{2}^{(k-i-1)}{\circ}{\mathscr{S}}_{2}^{i+1},$
(D.34)

with
${\mathfrak{a}}_{3}(\theta):={\mathfrak{a}}_{1}(\theta)+\partial_{\theta}c_{0}(\theta)$,
in the second line

$\partial_{\theta}p_{1}{\circ}{\mathscr{S}}_{1}^{k}-\partial_{\theta}p_{2}{\circ}{\mathscr{S}}_{2}^{k}=\Delta_{\partial_{\theta}p_{1},\partial_{\theta}p_{2},W,k}+\rho\,{\mathfrak{a}}_{4}{\circ}{\mathscr{S}}_{1}^{k}\,{\mathfrak{f}}{\circ}A_{0}^{k},$
(D.35)

with
${\mathfrak{a}}_{4}(\theta):=\partial_{\theta}{\mathfrak{a}}_{1}(\theta)$, and
in the third line

$\displaystyle
p_{1}^{(n-k-1)}{\circ}{\mathscr{S}}_{1}^{k+1}-p_{2}^{(n-k-1)}{\circ}{\mathscr{S}}_{2}^{k+1}$
(D.36)