# Synchronization and averaging
in dynamical systems with fast/slow variables

Federico Bonetto1 and Guido Gentile2
1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332,
USA
2 Dipartimento di Matematica, Università Roma Tre, Roma, 00146, Italy
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(August 28, 2024)

###### Abstract

We study a family of dynamical systems obtained by coupling a chaotic (Anosov)
map on the two-dimensional torus – the _chaotic system_ – with the identity
map on the one-dimensional torus – the _neutral system_ – through a
dissipative interaction. We show that the two systems synchronize: the
trajectories evolve toward an attracting invariant manifold, and the full
dynamics is conjugated to its linearization around the invariant manifold.
When the interaction is small, the evolution of the neutral variable, that is
the variable which describes the neutral system, is very close to the
identity; hence the neutral variable appears as a _slow_ variable with respect
to variable which describes the chaotic system, and which is wherefore named
the _fast_ variable. We demonstrate that, seen on a suitably long time scale,
the slow variable effectively follows the solution of a deterministic
differential equation obtained by averaging over the fast variable.

###### Contents

  1. 1 Introduction
  2. 2 Model and Results
    1. 2.1 Basic Ingredients
    2. 2.2 The model
    3. 2.3 Synchronization
      1. 2.3.1 The invariant manifold
      2. 2.3.2 The linearized map and the conjugation
      3. 2.3.3 The physical measure
    4. 2.4 Averaging
      1. 2.4.1 Heuristic discussion
      2. 2.4.2 Synchronization in the scaling regime
      3. 2.4.3 The averaged map
      4. 2.4.4 Oscillations and deviations in the scaling regime
      5. 2.4.5 Summing up: convergence in square mean and in probability
      6. 2.4.6 Aftermath: continuous time
    5. 2.5 Content of the paper and strategy of the proof
  3. 3 Relation with previous works
    1. 3.1 Synchronization under stronger assumptions
    2. 3.2 Expanding maps
  4. 4 Extensions and generalizations
    1. 4.1 More general perturbations: fully coupled systems
      1. 4.1.1 More general Anosov diffeomorphisms
      2. 4.1.2 Bidirectional perturbations
      3. 4.1.3 Non-dissipative perturbations
    2. 4.2 Central Limit Theorem
    3. 4.3 Large Deviations
  5. 5 Applications to physical problems
    1. 5.1 Periodic orbits: Krylov-Bogolyubov theory
    2. 5.2 Climate models
    3. 5.3 Several coupled systems: the heat equation
  6. 6 Mapping to a simpler model
    1. 6.1 The invariant manifold: proof of Theorem 1
    2. 6.2 The invariant manifold in the scaling regime: proof of Lemma 2.21
    3. 6.3 The conjugation: proof of Theorem 2
      1. 6.3.1 The translated map
      2. 6.3.2 Dynamics toward the steady state
      3. 6.3.3 Existence and regularity of the conjugation
    4. 6.4 The inverse conjugation: proof of Corollary 2.18
    5. 6.5 The conjugation in the scaling regime: proof of Lemma 2.22
    6. 6.6 Physical measure: proof of Theorem 3
  7. 7 Averaging and deviations
    1. 7.1 The extended map
    2. 7.2 A correlation inequality
      1. 7.2.1 A simple example
      2. 7.2.2 The general inequality
    3. 7.3 Oscillations of the invariant manifold: proof of Theorem 4
    4. 7.4 Fluctuations of the linearized dynamics: proof of Lemma 2.33
    5. 7.5 Iterated products
    6. 7.6 Conjugation of the averaged dynamics: proof of Lemma 2.27
    7. 7.7 The averaged and the continuous time system: proof of Lemma 2.29
    8. 7.8 Deviations of the conjugation: proof of Theorem 5
      1. 7.8.1 The auxiliary map
      2. 7.8.2 A new correlation inequality
      3. 7.8.3 Deviations of the conjugation of the auxiliary map
      4. 7.8.4 Comparison between the translated map and the auxiliary map
      5. 7.8.5 Deviations of the conjugation I: proof of the bounds (2.42a)
      6. 7.8.6 Deviations of the conjugation II: proof of the bounds (2.42b)
    9. 7.9 Deviations of the inverse conjugation
    10. 7.10 Fluctuations of the dynamics: proof of Theorem 6
  8. 8 Convergence in probability
    1. 8.1 The probability of deviations: proof of Theorem 7
    2. 8.2 Continuous time: proof of Theorem 8
  9. A Decay of correlations for Hölder continuous functions
    1. A.1 Symbolic dynamics
    2. A.2 Correlation functions
    3. A.3 A correlation inequality: proof of Proposition 2.3
  10. B Bounds on the norms of the iterated products
    1. B.1 A first application: completion of the proof of Theorem 2
    2. B.2 Products independent of the fast variable: proof of Lemma 7.16
    3. B.3 Products depending on the fast variable: proof of Lemma 7.27
  11. C A correlation inequality involving the slow variable
    1. C.1 Some preliminary rewriting
    2. C.2 The new correlation inequality: proof of Proposition 7.30
  12. D Proof of some technical results
    1. D.1 Derivative of the auxiliary map: proof of Proposition 7.33
    2. D.2 Comparing the translated and auxiliary maps I: proof of Lemma 7.37
    3. D.3 Comparing the translated and auxiliary maps II: proof of Proposition 7.50
    4. D.4 Second derivative of the conjugation: proof of Lemma 7.52
  13. E Triple correlations: proof of (4.9)

##  1 Introduction

Synchronization in quasi-integrable systems is well known to occur in the
presence of dissipation; a typical example is the orbital resonance in
celestial mechanics [44]. On contrast, in chaotic systems, where trajectories
starting at close initial conditions tend to diverge from each other,
synchronization may appear as an unlikely phenomenon. Nonetheless, the
presence of negative Lypaunov exponents due to a dissipative coupling can
still produce synchronization [46].

One of the simplest models one can think of is obtained by coupling a chaotic
system, for instance an Anosov automorphism on the two-dimensional torus
$\mathds{T}^{2}$, such as the Arnold’s cat map, with a one-dimensional neutral
system through a dissipative perturbation which is unidirectional, that is
which affects only the motion of the neutral system – by ‘neutral system’ we
mean a system which just remains at rest in the absence of the interaction.
Such a model has been explicitly considered in ref. [30], under suitable
assumptions which simplify the analysis to a great extent (see Subsection 3.1
for further details). More complicated and realistic models can be easily
envisaged [48, 49, 8, 33, 3, 47], however the main advantage of the simple
model studied in ref. [30] is that its solution can be explicitly worked out
and studied in great detail, without resorting to numerical simulations or
heuristic arguments. What is found is that a two-dimensional invariant
manifold appears on which the dynamics is conjugated to that of the
unperturbed automorphism: as a consequence, the two systems synchronize
asymptotically, in the sense that they tend to realize a drive-response
configuration, with the originally neutral system slaved to follow the
dynamics of the chaotic system (see ref. [8] and references therein for an
introduction to the topics). The invariant manifold is no more than Hölder
continuous, but, with the hypotheses considered in ref. [30], its oscillations
are small, that is of the same order as the perturbation – a property which is
not expected to hold in general.

In the present paper we study a class of dynamical systems that include those
considered in ref. [30], and show that an invariant manifold exists in a more
general setting, and it is still the graph of a Hölder continuous function
over the two-dimensional torus. Moreover, we extend the analysis beyond the
perturbative regime, by requiring the coupling only to be dissipative in a
finite region. The oscillations of the invariant manifold may be rather large
in general, even in the perturbative regime, albeit large oscillations are
rare in the latter case, given that both the average and the variance of the
function whose graph describes the invariant manifold is of the order of the
perturbation.

We also provide a detailed description of the dynamics away from the invariant
manifold, by demonstrating that it is conjugated to its linearization around
the invariant manifold. In particular, the invariant manifold is proved to be
an attractor. The conjugation too, in general, is no more than Hölder
continuous. In the perturbative regime, also the average deviations of the
conjugating function are found to be of the order of the perturbation, despite
the fact that deviations of order 1 are possible. The results discussed above
yield, as a byproduct, that any system in the class we consider admits a
unique physical measure given by the lift to the invariant manifold of the
normalized Lebesgue measure of $\mathds{T}^{2}$. This measure is exponentially
mixing, with mixing rate limited by the low regularity of the invariant
manifold.

When the perturbation is very small, two time scales naturally appear in the
evolution: the _fast_ time scale of the chaotic dynamics on $\mathds{T}^{2}$
as opposed to the _slow_ time scale of the neutral system, whose evolution is
driven only by the perturbation. The study of systems with fast and slow
variables is well established for quasi-integrable systems, where the motion
of the slow variable is close to periodic, and probably originated with
Lagrange’s analysis of the secular variations of the orbital elements of
planets [40]. In quasi-integrable systems, the slow variable, for a very long
time, only feels the average over one period of its interaction with the fast
variables. Thus, to study the drift of the slow variable, one applies the so-
called method of averaging [34, 5, 53, 50]: once the oscillations of the fast
variables have been integrated out, the approximate solution one finds
provides, in general, a reliable description of the dynamics up to a time
which is inversely proportional to the slow time scale; to make the analysis
rigorous, as the next step, one has to control the corrections.

In a similar spirit, we investigate the _scaling regime_ of the dynamics of
the systems we are considering, that is we fix a finite time $t$ and study
what happens, when the size $\rho$ of the perturbation is very small, after
$k=\lfloor t/\rho\rfloor$ iterations of the map. We find that, in this regime,
the evolution of the slow variable becomes essentially independent of the
dynamics on the torus, and is effectively described by the solution of a
suitable ordinary differential equation. The differential equation is
essentially the continuous limit of the original map, with the function of the
fast variables describing the interaction replaced by its average on the
torus. More precisely we show that, given an initial condition for the slow
variable, and taking a random initial condition for the fast variable, the
probability of seeing a sizable deviation from the deterministic averaged
evolution is of the order of the perturbation. Because of the presence of
dissipation, which makes the trajectories to evolve toward the invariant
manifold, the probability of such deviations remains small along the full
trajectory up to an infinite time. These results are related to the study in
refs. [21, 41, 23] on a similar class of models, in which the system described
by the slow variable is coupled, through a more general and not necessarily
uniformly dissipative perturbation, with an expanding circle map. On the other
hand, for dissipative interactions, the presence of both stable and unstable
directions for the fast chaotic dynamics on $\mathds{T}^{2}$ makes our study
more general – see also Subsection 3.2 for a more detailed comparison.

Of course, the fact that the neutral system does not influence the chaotic
evolution on $\mathds{T}^{2}$ makes the analysis easier. Notwithstanding this
simplification, we think that the model we study here contains most of the
relevant features to control also the statistical properties over long time
scales of models with more general couplings (see Section 4). At the same
time, it has the advantage of being well suited for explicit, direct
computations, and eliminating details which would introduce technical
complications without really adding anything to the underlying physics.
Therefore, in our opinion, the model represents a first step toward a full
mathematical understanding of the problem, before considering more realistic
situations.

Models as those considered above have been widely studied in the literature
also as a preliminary step of the more ambitious program of deriving
rigorously the heat equation from the microscopic equations of motions. In
this perspective, as pointed out in ref. [41], what one would really like to
investigate is the case of several chaotic systems coupled with an equal
number of neutral systems in a local manner and weakly interacting with each
other, and look for results which are uniform in the number of systems. One is
ultimately interested in taking the hydrodynamical limit, that is considering
infinitely many coupled systems obtained by a suitable scaling limit; a model
of this kind, with a different approach with respect to ours, has been studied
in ref. [12], where a diffusion equation for the macroscopic energy is derived
starting from the microscopic dynamics. For further comments on this line of
research we refer to Subsection 5.3.

##  2 Model and Results

In this section, first, we introduce the basic ingredients that will be used
in the rest of the paper: automorphisms of $\mathds{T}^{2}$, regularity norms
and relative Banach spaces, and correlation functions. Then, we give the
formal definition of the model we will study and present our main results,
referring to Sections 6 and 7 – and the Appendices – for the proofs.

###  2.1 Basic Ingredients

Let $\mathds{T}:=\mathds{R}/2\pi\mathds{Z}$ and consider an Anosov
automorphism $A_{0}$ of $\mathds{T}^{2}$ [2, 13, 24] such as Arnold’s cat map
[4]. Call $\lambda_{\pm}$ the eigenvalues and $v_{\pm}$ the eigenvectors of
$A_{0}$, with $|v_{\pm}|=1$, $\lambda_{+}>1>\lambda_{-}$ and
$\lambda_{-}\lambda_{+}=1$, and set $\lambda=\lambda_{+}$.

Let $\Omega:={\mathcal{U}}\times\mathds{T}^{2}$, where ${\mathcal{U}}$ is a
non-empty closed interval of either $\mathds{T}$ or $\mathds{R}$. For any
function $f\\!:\Omega\to\mathds{R}$ and any $\varphi\in{\mathcal{U}}$, set

$\langle f\rangle(\varphi):=\langle
f(\varphi,\cdot)\rangle:=\int_{\mathds{T}^{2}}f(\varphi,\psi)\,m_{0}({\rm
d}\psi),\qquad\tilde{f}(\varphi,\psi):=f(\varphi,\psi)-\langle
f\rangle(\varphi),$ (2.1)

where $m_{0}({\rm d}\psi):={\rm d}\psi/(2\pi)^{2}$ is the normalized Lebesgue
measure on $\mathds{T}^{2}$.

Consider the supremum norm

$\|f\|_{\infty}:=\sup_{(\varphi,\psi)\in\Omega}|f(\varphi,\psi)|,$ (2.2)

and let ${\mathscr{B}}(\Omega,\mathds{R})$ denote the Banach space of the
bounded continuous functions $f$ equipped with the norm $\|f\|_{\infty}$. For
$\alpha\in(0,1]$, consider the Hölder seminorm

$|f|_{\alpha}:=\sup_{(\varphi,\psi),(\varphi,\psi^{\prime})\in\Omega}\frac{|f(\varphi,\psi^{\prime})-f(\varphi,\psi)|}{|\psi^{\prime}-\psi|^{\alpha}}$
(2.3)

and the two directional Hölder seminorms

$|f|_{\alpha}^{\pm}:=\sup_{(\varphi,\psi)\in\Omega}\sup_{x\in\mathds{R}}\frac{|f(\varphi,\psi+xv_{\pm})-f(\varphi,\psi)|}{|x|^{\alpha}},$
(2.4)

that satisfy the inequalities

$|f|_{\alpha}^{\pm}\leq|f|_{\alpha}\leq|f|_{\alpha}^{+}+|f|_{\alpha}^{-},\qquad\qquad\|f\|_{\infty}\leq
c_{\alpha}|f|_{\alpha}+\|\langle f\rangle\|_{\infty},$ (2.5)

with $c_{\alpha}:=(\pi\sqrt{2})^{\alpha}$. Introduce also the norms

$\|f\|_{\alpha_{-},\alpha_{+}}:=\|f\|_{\infty}+\alpha_{-}|f|_{\alpha_{-}}^{-}+\alpha_{+}|f|_{\alpha_{+}}^{+},\qquad\|f\|_{\alpha}^{-}:=\|f\|_{\alpha,0},\qquad\|f\|_{\alpha}^{+}:=\|f\|_{0,\alpha},$
(2.6)

and let ${\mathscr{B}}^{*}_{\alpha_{-},\alpha_{+}}\\!(\Omega,\mathds{R})$,
${\mathscr{B}}_{\alpha}^{+}(\Omega,\mathds{R})$ and
${\mathscr{B}}_{\alpha}^{-}(\Omega,\mathds{R})$ denote the anisotropic Banach
spaces of the functions defined on $\Omega$ equipped with the norms
$\|\cdot\|_{\alpha_{-},\alpha_{+}}$, $\|\cdot\|_{\alpha}^{+}$ and
$\|\cdot\|_{\alpha}^{-}$, respectively. If $\alpha_{-}=\alpha_{+}=\alpha$ the
norm $\|\cdot\|_{\alpha,\alpha}$ is equivalent to the norm
$\|\cdot\|_{\alpha}:=\|\cdot\|_{\infty}+\alpha|\cdot|_{\alpha}$ of the
$\alpha$-Hölder continuous functions, so that
${\mathscr{B}}^{*}_{\alpha,\alpha}(\Omega,\mathds{R})={\mathscr{B}}_{\alpha}(\Omega,\mathds{R})$,
where ${\mathscr{B}}_{\alpha}(\Omega,\mathds{R})$ is the Banach space defined
by the norm $\|\cdot\|_{\alpha}$; on the other hand, using anisotropic Banach
spaces allows to treat differently the stable and unstable manifolds [7, 19],
and this will be exploited in what follows. Finally, observe that
${\mathscr{B}}_{0}(\Omega,\mathds{R})={\mathscr{B}}(\Omega,\mathds{R})$.

It is easy to see that

$|fg|_{\alpha}\leq\|f\|_{\infty}|g|_{\alpha}+|f|_{\alpha}\|g\|_{\infty},\qquad\qquad|fg|_{\alpha}^{\pm}\leq\|f\|_{\infty}|g|_{\alpha}^{\pm}+|f|_{\alpha}^{\pm}\|g\|_{\infty},$
(2.7)

and

$\|fg\|_{\alpha}\leq\|f\|_{\alpha}\|g\|_{\alpha},\qquad\|fg\|_{\alpha_{-},\alpha_{+}}\leq\|f\|_{\alpha_{-},\alpha_{+}}\|g\|_{\alpha_{-},\alpha_{+}}.$
(2.8)

For $k\geq 1$, define also ${\mathscr{B}}_{\alpha,k}(\Omega,\mathds{R})$ as
the Banach space of the functions $f\\!:\Omega\to\mathds{R}$ which are
$k$-times continuously differentiable in the first variable, that is in the
variable $\varphi\in{\mathcal{U}}$, and such that the first $k-1$ derivatives
are $\alpha$-Hölder continuous in the second one, that is in the variable
$\psi\in\mathds{T}^{2}$, equipped with the norm

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|f|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k}:=\sum_{n=0}^{k-1}\|\partial_{\varphi}^{n}f\|_{\alpha}+\|\partial_{\varphi}^{k}f\|_{\infty}=\sum_{n=0}^{k}\|\partial_{\varphi}^{n}f\|_{\infty}+\alpha\sum_{n=0}^{k-1}|\partial_{\varphi}^{n}f|_{\alpha}\,,$
(2.9)

where $\partial_{\varphi}$ denotes the derivative with respect to the first
variable. Similarly define the norms
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\cdot|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k}^{+}$
and
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\cdot|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k}^{-}$,
as in (2.9) with $|\cdot|_{\alpha}$ replaced with $|\cdot|_{\alpha}^{+}$ and
$|\cdot|_{\alpha}^{-}$, and call the respective Banach spaces
${\mathscr{B}}_{\alpha,k}^{+}(\Omega,\mathds{R})$ and
${\mathscr{B}}_{\alpha,k}^{-}(\Omega,\mathds{R})$. As in (2.8) we get

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|fg|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k}\leq|{\hskip-0.85355pt}|{\hskip-0.85355pt}|f|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k},\qquad|{\hskip-0.85355pt}|{\hskip-0.85355pt}|fg|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k}^{\pm}\leq|{\hskip-0.85355pt}|{\hskip-0.85355pt}|f|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k}^{\pm}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|g|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,k}^{\pm}.$
(2.10)

######  Remark 2.1.

_In the following we also consider sets of the form_

${\mathcal{A}}=\\{(\varphi,\psi):a_{-}(\psi)\leq\varphi\leq
a_{+}(\psi)\\}\subset\mathds{R}\times\mathds{T}^{2},$ (2.11)

_where $a_{\pm}\\!:\mathds{T}^{2}\to\mathds{R}$ are Hölder continuous
functions. All the definitions of the norms given above extend naturally if
the set $\Omega$ is replaced with any other closed subset of
${\mathcal{A}}\subset\mathds{R}\times\mathds{T}^{2}$ of the form (2.11). We
only need to take the supremum over $(\varphi,\psi)\in{\mathcal{A}}$ and
replace (2.3) and (2.4) with_

$\displaystyle|f|_{\alpha}$
$\displaystyle:=\sup_{(\varphi,\psi),(\varphi,\psi^{\prime})\in{\mathcal{A}}}\frac{|f(\varphi,\psi^{\prime})-f(\varphi,\psi)|}{|\psi^{\prime}-\psi|^{\alpha}}$
$\displaystyle|f|_{\alpha}^{\pm}$
$\displaystyle:=\sup_{(\varphi,\psi)\in{\mathcal{A}}}\sup_{\genfrac{}{}{0.0pt}{1}{x\in\mathds{R}}{(\varphi,\psi+xv_{\pm})\in{\mathcal{A}}}}\frac{|f(\varphi,\psi+xv_{\pm})-f(\varphi,\psi)|}{|x|^{\alpha}},$

_respectively. This allows us to define the corresponding Banach spaces in the
same way as before, with the set ${\mathcal{A}}$ instead of $\Omega$. _

In order not to introduce further symbols, we use the notation
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|f|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,k}$
also to denote the $C^{k}$-norm of any function $f$ depending only on the
first variable $\varphi$. We thus identify $C^{k}({\mathcal{D}},\mathds{R})$,
for any given subset ${\mathcal{D}}\subseteq\mathds{R}$, with the subspace of
${\mathscr{B}}_{0,k}({\mathcal{D}}\times\mathds{T}^{2},\mathds{R})$ of the
functions independent of $\psi$.

For clarity sake, we call
${\mathfrak{B}}^{*}_{\alpha_{+},\alpha_{-}}\\!(\mathds{T}^{2},\mathds{R})$ the
subspace of ${\mathscr{B}}^{*}_{\alpha_{+},\alpha_{-}}\\!(\Omega,\mathds{R})$
of functions that do not depend on $\varphi$, and similarly for
${\mathfrak{B}}_{\alpha}^{\pm}(\mathds{T}^{2},\mathds{R})$,
${\mathfrak{B}}_{\alpha}(\mathds{T}^{2},\mathds{R})$ and
${\mathfrak{B}}(\mathds{T}^{2},\mathds{R})$. For such functions, the seminorms
$|\cdot|_{\alpha}^{\pm}$ and the norms $\|\cdot\|_{\infty}$,
$\|\cdot\|_{\alpha_{,}\alpha_{+}}$ and $\|\cdot\|_{\alpha}^{\pm}$ are defined
as previously, with the supremum taken over $\psi\in\mathds{T}^{2}$ only.

######  Remark 2.2.

_The behavior of the seminorms ( 2.3) and (2.4) for
$f\in{\mathfrak{B}}_{\alpha}(\mathds{T}^{2},\mathds{R})$ under the action of
$A_{0}^{n}$, for $n\in\mathds{Z}$, is given by_

$|f{\circ}A_{0}^{n}|_{\alpha}\leq\lambda^{\alpha|n|}|f|_{\alpha}\,,\qquad|f{\circ}A_{0}^{n}|_{\alpha}^{+}\leq\lambda^{\alpha
n}|f|_{\alpha}^{+}\,,\qquad|f{\circ}A_{0}^{n}|_{\alpha}^{-}\leq\lambda^{-\alpha
n}|f|_{\alpha}^{-}\,.$ (2.12)

_If the functions $g_{0,+},\ldots,g_{n-1,+}$ are in
${\mathfrak{B}}_{\alpha}^{+}(\mathds{T}^{2},\mathds{R})$, for some
$\alpha\in(0,1]$, then_

$\Bigl{|}\prod_{i=0}^{n-1}g_{i,+}{\circ}A_{0}^{-i}\Bigr{|}_{\alpha}^{+}\leq\sum_{i=0}^{n-1}\lambda^{-\alpha
i}|g_{i,+}|_{\alpha}^{+}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\|g_{i,+}\|_{\infty}.$ (2.13)

_Analogously, one has_

$\Bigl{|}\prod_{i=0}^{n-1}g_{i,-}{\circ}A_{0}^{i}\Bigr{|}_{\alpha}^{-}\leq\sum_{i=0}^{n-1}\lambda^{-\alpha
i}|g_{i,-}|_{\alpha}^{-}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\|g_{i,-}\|_{\infty},$ (2.14)

_if the functions $g_{0,-},\ldots,g_{n-1,-}$ are in
${\mathfrak{B}}_{\alpha}^{-}(\mathds{T}^{2},\mathds{R})$, for some
$\alpha\in(0,1]$. _

A crucial role in our analysis is played by the correlation functions and
their decay due to the hyperbolicity of $A_{0}$. In Appendix A.3 we prove the
following estimate.

######  Proposition 2.3.

Let the functions $g_{+}$ and $g_{-}$ be, respectively, in
${\mathfrak{B}}_{\alpha}^{+}(\mathds{T}^{2},\mathds{R})$ and in
${\mathfrak{B}}_{\alpha}^{-}(\mathds{T}^{2},\mathds{R})$, for some
$\alpha\in(0,1]$. Then for all $n\in\mathds{N}$ one has

$\left|\left\langle g_{+}\,g_{-}{\circ}A_{0}^{n}\right\rangle-\langle
g_{+}\rangle\langle g_{-}\rangle\right|\leq C(1+\alpha n)\lambda^{-\alpha
n}\|\tilde{g}_{+}\|_{\alpha}^{+}\|\tilde{g}_{-}\|_{\alpha}^{-}\,,$

for a suitable positive constant $C$ independent of $n$, $\alpha$, $g_{-}$ and
$g_{+}$.

######  Remark 2.4.

_Proposition 2.3 implies that, under the same assumptions, for every
$\alpha^{\prime}<\alpha$ one has_

$\left|\left\langle g_{+}\,g_{-}{\circ}A_{0}^{n}\right\rangle-\langle
g_{+}\rangle\langle g_{-}\rangle\right|\leq
C^{\prime}\lambda^{-\alpha^{\prime}n}\|\tilde{g}_{+}\|_{\alpha}^{+}\|\tilde{g}_{-}\|_{\alpha}^{-},$

_with the positive constant $C^{\prime}$ depending only on $\alpha^{\prime}$.
_

###  2.2 The model

We consider the dynamical system defined by the map ${\mathscr{S}}$ on
$\mathds{T}\\!\times\\!\mathds{T}^{2}$ given by

${\mathscr{S}}(\varphi,\psi)=({\mathscr{S}}_{\varphi}(\varphi,\psi),{\mathscr{S}}_{\psi}(\varphi,\psi))=(G(\varphi,\psi),A_{0}\psi),\qquad\qquad
G(\varphi,\psi):=\varphi+F(\varphi,\psi)\,,$ (2.15)

with
$F\in{\mathscr{B}}_{\alpha_{0},6}(\mathds{T}\times\mathds{T}^{2},\mathds{R})$,
for some $\alpha_{0}\in(0,1]$, and assume ${\mathscr{S}}$ to satisfy the
following hypotheses.

######  Remark 2.5.

_The assumption of Hölder continuity of the map in the variable $\psi$ is very
natural, since it cannot be weakened and, at the same time, requiring stronger
regularity would not provide stronger results (see also Remarks 2.19 and 6.3
below). On contrast, not only the regularity in the variable $\varphi$ may not
be optimal, but some of the very same results proved along the paper require
much less regularity (see for instance Remark 2.15); on the other hand, as far
as the variable $\varphi$ is concerned, higher regularity of the map does
yield higher regularity of the long time behavior of the dynamics (see Remark
2.16). _

######  Hypothesis 1.

There exists a non-empty closed interval ${\mathcal{U}}\subsetneq\mathds{T}$
such that $\partial_{\varphi}{\mathscr{S}}_{\varphi}(\varphi,\psi)>0$ for all
$(\varphi,\psi)\in\Omega:={\mathcal{U}}\\!\times\\!\mathds{T}^{2}$.

######  Hypothesis 2.

If, after a suitable parametrization of $\mathds{T}$, one writes
${\mathcal{U}}=[\phi_{m},\phi_{M}]$, with $\phi_{m}<\phi_{M}$, then
${\mathscr{S}}_{\varphi}(\phi_{m},\psi)>\phi_{m}$ and
${\mathscr{S}}_{\varphi}(\phi_{M},\psi)<\phi_{M}$ for all
$\psi\in\mathds{T}^{2}$.

Hypotheses 1 and 2 imply that for every $\psi\in\mathds{T}^{2}$ there is a
unique point $S(\psi)\in{\rm int}\,{\mathcal{U}}$ with $F(S(\psi),\psi)=0$.
Set

$S_{m}:=\inf_{\psi\in\mathds{T}^{2}}S(\psi),\qquad\qquad
S_{M}:=\sup_{\psi\in\mathds{T}^{2}}S(\psi),$ (2.16)

and define

$\Lambda:=[S_{m},S_{M}]\times\mathds{T}^{2},\qquad\qquad\Gamma:=-\sup_{(\varphi,\psi)\in\Lambda}\partial_{\varphi}F(\varphi,\psi).$
(2.17)

######  Hypothesis 3.

One has $\Gamma>0$.

######  Remark 2.6.

_A simple example of a map satisfying Hypotheses 1–3, with, say,
${\mathcal{U}}=[-\pi/4,\pi/4]$, is obtained by taking
$F(\varphi,\psi)=-\sin(\varphi-g(\psi))$, with
$g(\psi)=(\pi/8)\,\cos(\psi_{1}-\psi_{2})$. Indeed, in this case we find
$S(\psi)=g(\psi)$, so that $\Lambda=[-\pi/8,\pi/8]$ and $\Gamma=\cos(\pi/4)$.
_

######  Remark 2.7.

_Hypothesis 1 yields that ${\mathscr{S}}$ is injective on $\Omega$ and that
$\Gamma<1$. Moreover, as a consequence of both Hypotheses 1 and 2, one has
${\mathscr{S}}(\Omega)\subsetneq{\rm int}\,\Omega$ and
$S(\psi)\in(\phi_{m},\phi_{M})$ for all $\psi\in\mathds{T}^{2}$, and hence
$[S_{m},S_{M}]\subsetneq(\phi_{m},\phi_{M})$. _

######  Remark 2.8.

_Throughout the paper, for any map ${\mathscr{S}}$, the notation
${\mathscr{S}}^{n}$ means
${\mathscr{S}}{\circ}{\mathscr{S}}^{n-1}={\mathscr{S}}{\circ}\ldots{\circ}{\mathscr{S}}$
, that is the composition of ${\mathscr{S}}$ with itself $n$ times. _

The following lemma lists a few immediate consequences of Hypotheses 1–3.

######  Lemma 2.9.

If ${\mathscr{S}}$ satisfies Hypotheses 1–3, then the following properties
hold:

  1. 1.

one has $F(\varphi,\psi)>0$ for $\varphi<S_{m}$, while $F(\varphi,\psi)<0$ for
$\varphi>S_{M}$;

  2. 2.

one has $\partial_{\varphi}F(\varphi,\psi)>-1$ for $(\varphi,\psi)\in\Omega$
and hence $-1<\partial_{\varphi}F(\varphi,\psi)\leq-\Gamma$ for
$(\varphi,\psi)\in\Lambda$;

  3. 3.

for any $r>0$ there exists $N_{r}\in\mathds{N}$ such that
${\mathscr{S}}^{N_{r}}(\Omega)\subset\Lambda_{r}:=[S_{m}-r,S_{M}+r]\times\mathds{T}^{2}$,
with

$N_{r}\leq\frac{\displaystyle{\max\\{\phi_{M}-S_{M},S_{m}-\phi_{m}\\}}}{\displaystyle{\inf_{\Omega\backslash\Lambda_{r}}|F(\varphi,\psi)|}}\,;\vspace{-.2cm}$
(2.18)

  4. 4.

for any $\Gamma^{\prime}\in(0,\Gamma)$ there exists $r=r(\Gamma^{\prime})$
such that $\partial_{\varphi}F(\varphi,\psi)\leq-\Gamma^{\prime}$ for all
$(\varphi,\psi)\in\Lambda_{r}$;

  5. 5.

the set $\Lambda$ is positively invariant under ${\mathscr{S}}$ and is
attracting for ${\mathscr{S}}$ on $\Omega$;

  6. 6.

there exists a unique $\overline{\varphi}\in(S_{m},S_{M})$ such that $\langle
F(\overline{\varphi},\cdot)\rangle=0$.

######  Remark 2.10.

_Due to property 6 in Lemma 2.9, without loss of generality, we may and do
choose the parametrization in Hypothesis 2 in such a way that
$\overline{\varphi}=0$. _

######  Remark 2.11.

_By Remark 2.10 we can write_

$F(\varphi,\psi)=\beta(\psi)-\nu(\psi)\,\varphi+\delta(\varphi,\psi)\,\varphi^{2},$
(2.19)

_where $\langle\beta\rangle=0$. Conversely, assume that $F$ has the form_

$F(\varphi,\psi)=\beta(\psi)-\nu(\psi)\,\varphi+\delta(\psi)\,\varphi^{2},$
(2.20)

_with $\langle\beta\rangle=0$ and
$4\|\delta\|_{\infty}\|\beta\|_{\infty}<\nu_{0}^{2}$, if
$\nu_{0}:=\inf_{\psi\in\mathds{T}^{2}}|\nu(\psi)|$. It is easy to see that
(2.15) defined by such an $F$ satisfies Hypotheses 1–3 with
$\phi_{M}=\|\beta\|_{\infty}/2\nu_{0}$ and $\phi_{m}=-\phi_{M}$. _

######  Remark 2.12.

_Although we defined ${\mathscr{S}}$ as a map on
$\mathds{T}\times\mathds{T}^{2}$, since
${\mathscr{S}}(\Omega)\subsetneq\Omega$, in the following we will only be
interested in the action of ${\mathscr{S}}$ on $\Omega$ and identify $\Omega$
as a subset of $\mathds{R}\times\mathds{T}^{2}$. For technical reasons, we
will also need to extend the map ${\mathscr{S}}|_{\Omega}$ to a map
${\mathscr{S}}_{\rm ext}$ on $\Omega_{\rm ext}=\mathcal{U}_{\rm
ext}\times\mathds{T}^{2}$, for some closed interval ${\mathcal{U}}_{\rm ext}$
such that ${\mathcal{U}}\subset{\mathcal{U}}_{\rm ext}\subset\mathds{R}$, in
such a way that Hypotheses 1 to 3 remain valid (we refer to Subsection 7.1 for
further details). _

We call $\varphi$ the _slow variable_ and $\psi$ the _chaotic_ or _fast
variable_ – such a terminology is motivated by the fact that we are mainly
interested in the limit of small $\Gamma$, where the neutral variable
$\varphi$ moves slowly with respect to the chaotic variable $\psi$ describing
the hyperbolic system.

###  2.3 Synchronization

####  2.3.1 The invariant manifold

Observe that, if in (2.15) we replace the automorphism $A_{0}$ with the
identity $\mathds{1}$, that is if we consider the dynamics generated by
${\mathscr{S}}_{\mathds{1}}(\varphi,\psi)=(\varphi+F(\varphi,\psi),\psi)$ with
$F$ still satisfying Hypotheses 1–3, then
${\mathcal{W}}_{\mathds{1}}=\\{(S(\psi),\psi):\psi\in\mathds{T}^{2}\\}$ is an
invariant manifold in the sense that
${\mathscr{S}}_{\mathds{1}}({\mathcal{W}}_{\mathds{1}})={\mathcal{W}}_{\mathds{1}}$.
It is natural to ask whether a similar property remains true for
${\mathscr{S}}$ notwithstanding the chaotic nature of the evolution generated
by $A_{0}$ on $\mathds{T}^{2}$. More precisely we say that a manifold
${\mathcal{W}}=\\{(W(\psi),\psi):\psi\in\mathds{T}^{2}\\}$ is invariant for
${\mathscr{S}}$ in (2.15) if we have

${\mathscr{S}}(W(\psi),\psi)=(W(A_{0}\psi),A_{0}\psi)$ (2.21)

for every $\psi\in\mathds{T}^{2}$. This also means that on ${\mathcal{W}}$ the
dynamics generated by ${\mathscr{S}}$ is conjugated to the dynamics generated
by to the map $\psi\mapsto A_{0}\psi$.

In Subsection 6.1 we prove the following result.

######  Theorem 1 (Synchronization).

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.15)
satisfying Hypotheses 1–3. There exists a unique invariant manifold
${\mathcal{W}}=\\{(W(\psi),\psi):\psi\in\mathds{T}^{2}\\}\subset\Lambda$ for
the map ${\mathscr{S}}\\!$, with
$W\in{\mathfrak{B}}^{*}_{\alpha_{-},\alpha_{+}}\\!(\mathds{T}^{2},\mathds{R})$
for suitable $\alpha_{-},\alpha_{+}\in(0,\alpha_{0}]$.

######  Remark 2.13.

_The main effect of the hyperbolicity of $A_{0}$ is that, in general, the
manifold ${\mathcal{W}}$ is only Hölder continuous in $\psi$ even if we take
$F$ very smooth in $\psi$. Moreover $W$ is, in general, of order 1 in
$\|F\|_{\infty}$ even if $F$ is very close to 0, its maximum size depending
mainly on $S(\psi)$ – see also Remark 6.4 below. On the other hand, the
existence of the manifold ${\mathcal{W}}$ (and of the conjugation
${\mathscr{H}}$ discussed later) remains true if we assume that $F$ is only
bounded in $\psi$. _

####  2.3.2 The linearized map and the conjugation

To analyze the evolution generated by ${\mathscr{S}}$ outside ${\mathcal{W}}$
we can try to conjugate it with its linearization around ${\mathcal{W}}$, that
is with the simpler system ${\mathscr{S}}_{0}$ given by

${\mathscr{S}}_{0}(\eta,\psi)=(\kappa(\psi)\eta,A_{0}\psi),\qquad\qquad\kappa(\psi):=1+\partial_{\varphi}F(W(\psi),\psi),$
(2.22)

where, by Hypothesis 3, one has $\kappa(\psi)\in(0,1-\Gamma)$ for all
$\psi\in\mathds{T}^{2}$. This means that we look for a function
${\mathscr{H}}$ such that

${\mathscr{H}}{\circ}{\mathscr{S}}={\mathscr{S}}_{0}{\circ}{\mathscr{H}}.$
(2.23)

A function ${\mathscr{H}}$ that satisfies (2.23) is called a _conjugating
function_ – or simply _conjugation_.

######  Remark 2.14.

_Since ${\mathscr{S}}_{0}$ is linear in $\eta$, it is easy to see that if
${\mathscr{H}}$ is a solution to (2.23) then, for any $a\not=0$, also
${\mathscr{H}}(a\varphi,A_{0}\psi)$ is a solution. Thus we say that
${\mathscr{H}}$ is _the_ conjugation if it solves (2.23) and can be written
as_

${\mathscr{H}}(\varphi,\psi)=({\mathcal{H}}(\varphi,\psi),\psi),$ (2.24)

_with $\partial_{\varphi}{\mathcal{H}}(0,\psi)=1$ for every
$\psi\in\mathds{T}^{2}$. _

In Subsection 6.3 we prove the following result on the conjugation.

######  Theorem 2 (Conjugation).

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.15)
satisfying Hypotheses 1–3. There exist a set
$\Omega_{0}\subset\mathds{R}\times\mathds{T}^{2}$ and a function
${\mathscr{H}}\\!:\Omega\to\Omega_{0}$ that conjugates ${\mathscr{S}}$ via
(2.23) to ${\mathscr{S}}_{0}$ given by (2.22). Moreover ${\mathscr{H}}$ can be
written as in (2.24) with
${\mathcal{H}}\in{\mathscr{B}}_{\alpha_{*},2}(\Omega,\mathds{R})$ for a
suitable $\alpha_{*}\in(0,\min\\{\alpha_{-},\alpha_{+}\\}]$ and
$\alpha_{-},\alpha_{+}$ as in Theorem 1.

######  Remark 2.15.

_For Theorems 1 and 2 – and for the forthcoming Theorems 3 and 4 as well – to
be valid it would be enough to assume
$F\in{\mathscr{B}}_{\alpha_{0},3}(\Omega,\mathds{R})$. Higher regularity will
be needed to prove Theorem 5, which in turn plays an essential role in the
proof of Theorems 6 and 7. _

######  Remark 2.16.

_By looking at the proof of Theorem 2 (see Subsection 6.3.3 and Appendix B.1),
one may infer that if $F\in{\mathscr{B}}_{\alpha_{0},k}(\Omega,\mathds{R})$
for some $k\in\mathds{N}$, then
${\mathcal{H}}\in{\mathscr{B}}_{\alpha_{*},k-1}(\Omega,\mathds{R})$. In
particular, under the assumption that $F$ is in
${\mathscr{B}}_{\alpha_{0},6}$, the function ${\mathcal{H}}$ can be proved to
be $C^{5}$ in the slow variable; however, in the following, we do not need
more regularity of the conjugation than that stated in Theorem 2. _

######  Remark 2.17.

_Theorem 2 implies that, for any initial datum $(\varphi,\psi)\in\Omega$, the
evolution generated by ${\mathscr{S}}$ leads towards the invariant manifold
${\mathcal{W}}$. Therefore, the invariant manifold is a global attractor for
the dynamical system $(\Omega,{\mathscr{S}})$. _

From the proof of Theorem 2 it is easy to see that the function
${\mathscr{H}}$ is invertible; indeed, the following result is proved in
Subsection 6.4.

######  Corollary 2.18.

The map ${\mathscr{H}}$ in Theorem 2 is invertible, and its inverse
${\mathscr{H}}^{-1}\\!:\Omega_{0}\to\Omega$, which satisfies

${\mathscr{H}}^{-1}{\circ}{\mathscr{S}}_{0}={\mathscr{S}}{\circ}{\mathscr{H}}^{-1}\,,$

can be written as
${\mathscr{H}}^{-1}(\eta,\psi)=({\mathcal{L}}(\eta,\psi),\psi)$, with
${\mathcal{L}}\in{\mathscr{B}}_{\alpha_{*},1}(\Omega_{0},\mathds{R})$.

######  Remark 2.19.

_As for $W$, the conjugation and its inverse are no more than Hölder
continuous even if $F$ is very smooth in $\psi$. From a technical point of
view, the analysis might be simplified by assuming slightly stronger
regularity conditions for $F$, such as the strong Hölder condition considered
in ref. [6]. Such a condition however would force us to restrict the analysis
to systems of the form (2.20) with more stringent bounds on the functions
$\beta$, $\nu$ and $\delta$ then those in Remark 2.11. _

The proofs of Theorem 2 and of Corollary 2.18 in Subsection 6.3 and in
Subsection 6.4, respectively, show that ${\mathcal{H}}$ and ${\mathcal{L}}$
can be written as

${\mathcal{H}}(\varphi,\psi)=\varphi-W(\psi)+(\varphi-W(\psi))^{2}h(\varphi,\psi),\qquad{\mathcal{L}}(\eta,\psi)=W(\psi)+\eta+\eta^{2}l(\eta,\psi),$
(2.25)

with $h\\!\in\\!{\mathscr{B}}_{\alpha_{*},1}(\Omega,\mathds{R})$ and
$l\\!\in\\!{\mathscr{B}}_{\alpha_{*},1}(\Omega_{0},\mathds{R})$. These
representations will be useful in Subsection 2.4.4, when studying the
deviations of the conjugation.

####  2.3.3 The physical measure

Theorems 1 and 2 imply that, given any observable
${\mathcal{O}}\in{\mathscr{B}}_{\alpha,0}(\Omega,\mathds{R})$, one has

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}{\mathcal{O}}({\mathscr{S}}^{n}(\varphi_{0},\psi_{0}))=\int{\mathcal{O}}(W(\psi),\psi)\,m_{0}({\rm
d}\psi)=:\nu_{W}({\mathcal{O}})$

for almost every $(\varphi_{0},\psi_{0})\in\Omega$ with respect to the measure

$\nu_{0}({\rm d}\varphi{\rm d}\psi):=m_{0}({\rm d}\psi)\frac{{\rm
d}\varphi}{\phi_{M}-\phi_{m}}.$

Hence, $\nu_{W}$ is the unique physical measure for ${\mathscr{S}}$ on
$\Omega$. It is thus interesting to study the mixing property of $\nu_{W}$
with respect to $\nu_{0}$. The following result is proved in Subsection 6.6.

######  Theorem 3 (Mixing on the invariant manifold).

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.15)
satisfying Hypotheses 1–3. For any observables ${\mathcal{O}}_{1}$ and
${\mathcal{O}}_{2}$ in ${\mathscr{B}}_{\alpha_{*},1}(\Omega,\mathds{R})$, with
$\alpha_{*}$ as in Theorem 2, and any
${\lambda_{*}^{-1}}>\max\\{\lambda^{-\alpha_{*}},{1-\Gamma}\\}$, one has

$\left|\nu_{0}\left({\mathcal{O}}_{1}\,{\mathcal{O}}_{2}{\circ}{\mathscr{S}}^{n}\right)-\nu_{0}\left({\mathcal{O}}_{1}\right)\nu_{W}\left({\mathcal{O}}_{2}\right)\right|\leq
C\,{|{\hskip-0.85355pt}|{\hskip-0.85355pt}|{\mathcal{O}}_{1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{*},0}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|{\mathcal{O}}_{2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{*},1}}{\lambda_{*}^{-n}},$

where $C$ depends only on $\lambda_{*}$.

###  2.4 Averaging

We are interested in the long time evolution generated by the map
${\mathscr{S}}\\!$ in (2.15), when the component ${\mathscr{S}}_{\varphi}$ is
close to the identity. To this aim, we consider the family of functions
$F(\varphi,\psi)=\rho f(\varphi,\psi)$, with
$f\in{\mathscr{B}}_{\alpha_{0},6}(\mathds{T}\times\mathds{T}^{2},\mathds{R})$
and $\rho>0$ a parameter, and study the behaviour of the map

${\mathscr{S}}(\varphi,\psi)=(\varphi+\rho f(\varphi,\psi),A_{0}\psi)$ (2.26)

when $\rho$ is small. We also define

$\gamma:=-\sup_{(\varphi,\psi)\in\Lambda}\partial_{\varphi}f(\varphi,\psi),$
(2.27)

so that $\Gamma=\rho\,\gamma$ in (2.17).

######  Remark 2.20.

_We may and do assume, without loss of generality, that
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|f|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},6}=1$,
and hence, since
$\gamma<|{\hskip-0.85355pt}|{\hskip-0.85355pt}|f|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},6}$,
that $\gamma\in(0,1)$. For any fixed function
$f\in{\mathscr{B}}_{\alpha_{0},6}(\mathds{T}\times\mathds{T}^{2},\mathds{R})$
such that
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|f|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},6}=1$,
both Hypotheses 1 and 3 are automatically satisfied for $\rho$ small enough.
On the other hand, Hypothesis 1 requires $\rho$ not to be arbitrarily large
because we must have_

$\rho\,\partial_{\varphi}f(\varphi,\psi)>-1\qquad\forall(\varphi,\psi)\in\Omega.$
(2.28)

_Therefore, when considering a map ${\mathscr{S}}$ of the form 2.26, we
tacitly assume $\rho$ to be smaller than a a suitable value $\rho_{*}$,
depending on $f$, such that ${\mathscr{S}}$ satisfies Hypotheses 1 to 3 for
all $\rho\in(0,\rho_{*})$. _

We investigate the evolution generated by ${\mathscr{S}}$, as given in (2.26),
when $\rho\to 0^{+}$. To avoid such an evolution to become trivial, for given
initial conditions $(\varphi_{0},\psi_{0})$, we study the dynamics after a
linear _rescaling_ of time, that is we consider
${\mathscr{S}}^{k}(\varphi_{0},\psi_{0})$ taking $k=\lfloor t/\rho\rfloor$ for
$t$ fixed as $\rho\to 0^{+}$. We refer to the case with $\rho$ small as the
_scaling regime_ , even when studying the steady state of ${\mathscr{S}}$,
where ${k}$ does not explicitly appear.

####  2.4.1 Heuristic discussion

If $\rho$ is small enough we can take $k$ very large but still much smaller
than ${\rho^{-1}}$. Expanding ${\mathscr{S}}^{k}$ to first order in $\rho$ we
write

$({\mathscr{S}}^{k})_{\varphi}(\varphi,\psi)=\varphi+\rho\sum_{i=0}^{k}f(\varphi,A_{0}^{i}\psi)+{o(k\rho)\,.}$

Since $A_{0}$ is strongly mixing, we further obtain that, for most values of
$\psi_{0}$,

$({\mathscr{S}}^{k})_{\varphi}(\varphi,\psi)\simeq\varphi+k\rho\,\langle
f\rangle(\varphi)+o(k\rho)\,,$

where the right hand side has lost any dependence on $\psi$, at least at first
order in $\rho$. Calling $k\rho=t$, and writing
$\varphi(t):={\mathscr{S}}^{\lfloor t/\rho\rfloor}(\varphi,\psi)$ we can read
the last expression as

$\varphi(t)\simeq\varphi+\langle f\rangle(\varphi)\,t+{o(t)}\,.$

This propounds that, for $\rho$ small, the evolution $\varphi(t)$ that starts
from a given $\varphi$ and a randomly chosen $\psi$ is essentially independent
of $\psi$ and it agrees at first order in $t$ with the solution $\phi(t)$ of
the Cauchy problem

$\left\\{\begin{aligned} &\dot{\phi}=\langle f\rangle(\phi)\\\
&\phi(0)=\varphi\,.\end{aligned}\right.$ (2.29)

To see whether we can get a better agreement, we expand ${\mathscr{S}}^{k}$ to
second order in $\rho$ and find

$({\mathscr{S}}^{k})_{\varphi}(\varphi,\psi)=\varphi+\rho\sum_{i=0}^{k}f(\varphi,A_{0}^{i}\psi)+\rho^{2}\sum_{0\leq
i<j\leq
k}\partial_{\varphi}f(\varphi,A_{0}^{j}\psi)f(\varphi,A_{0}^{i}\psi)+{o(k^{2}\rho^{2})}\,,$

so that, for $\varphi(t)$ to remain close to $\phi(t)$ up to corrections
$o(t^{2})$, we need that

$\sum_{0\leq i<j\leq
k}\partial_{\varphi}f(\varphi,A_{0}^{j}\psi)f(\varphi_{0},A_{0}^{i}\psi)\simeq\frac{k^{2}}{2}\partial_{\varphi}\langle
f\rangle(\varphi)\langle f\rangle(\varphi),$

that is we need a strong form of decay of correlations for $A_{0}$.

This suggests that, on the correct time scale,
$({\mathscr{S}}^{k})_{\varphi}(\varphi,\psi)$ evolves according to a
differential equation involving only the average of $f$. This is a simple
instance of the idea of _averaging_ induced by the chaotic behavior of
$A_{0}$.

Clearly the argument above is only heuristic. In fact, extending the analysis
outlined above to all orders is likely to get too tangled and to require very
high regularity of the map. Moreover such an analysis is not suitable for
dealing with the case of arbitrary $t$, in particular for deriving results
uniform in $t$. Therefore, the heuristic argument only hints what to look for,
but, in order to obtain something rigorous, actually we follow a different
approach.

####  2.4.2 Synchronization in the scaling regime

The following two lemmas collect the implications of Theorems 1 and 2 and
their proofs for the dynamical system ${\mathscr{S}}$ in (2.26) with $\rho$
small – for the proofs see Subsections 6.2 and 6.5.

######  Lemma 2.21.

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Let
${\mathcal{W}}=\\{(W(\psi),\psi):\psi\in\mathds{T}^{2}\\}$ be the invariant
manifold for the map ${\mathscr{S}}$ as in Theorem 1. Then one has
$\alpha_{+}=O(1)$ and $\alpha_{-}=O(\rho)$, and, furthermore, both
$\|W\|_{\infty}$ and $|W|_{\alpha_{-}}^{-}$ are $O(1)$, while
$|W|_{\alpha_{+}}^{+}=O(\rho)$.

######  Lemma 2.22.

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Let ${\mathscr{H}}$ be the conjugation (2.24) for
the map ${\mathscr{S}}$ as in Theorem 2, and let $h$ and $l$ be defined as in
(2.25). Then one has $\alpha_{*}=O(\rho)$, while both
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|h|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{*},1}$
and
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|l|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{*},1}$
are $O(1)$ in $\rho$.

######  Remark 2.23.

_Lemma 2.21 shows that, when $\rho$ is small, the manifold ${\mathcal{W}}$
loses most of the smoothness of $f$ in the stable direction while maintaining
it in the unstable one. Moreover the manifold varies slowly in the unstable
direction. _

By using Lemma 2.21, the following corollary follows immediately from Theorem
3.

######  Corollary 2.24.

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. For any observables ${\mathcal{O}}_{1}$ and
${\mathcal{O}}_{2}$ in ${\mathscr{B}}_{\alpha_{*},1}$, with $\alpha_{*}$ as in
Theorem 2, and for all $n\in\mathds{N}$ one has

$\left|\nu_{0}\left({\mathcal{O}}_{1}\,{\mathcal{O}}_{2}{\circ}{\mathscr{S}}^{n}\right)-\nu_{0}\left({\mathcal{O}}_{1}\right)\nu_{W}\left({\mathcal{O}}_{2}\right)\right|\leq
C\,{|{\hskip-0.85355pt}|{\hskip-0.85355pt}|{\mathcal{O}}_{1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{*},0}}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|{\mathcal{O}}_{2}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{*},1}e^{-\xi\rho
n},$

where $C$ and $\xi$ are suitable constants not depending on $\rho$.

####  2.4.3 The averaged map

As a intermediate step toward a rigorous justification of the conclusions in
Subsection 2.4.1, together with the dynamical system described by (2.26) we
consider also the dynamical system given by the _averaged map_

$\overline{\mathscr{S}}(\varphi,\psi):=(\overline{G}(\varphi),A_{0}\psi)\,,\qquad\qquad\overline{G}(\varphi):=\varphi+\rho\overline{f}(\varphi),$
(2.30)

with

$\overline{f}(\varphi):=\langle f\rangle(\varphi),$ (2.31)

and its linearization

$\overline{\mathscr{S}}_{0}(\varphi,\psi):=(\overline{\mu}\,\varphi,A_{0}\psi),\qquad\qquad\overline{\mu}:=\partial_{\varphi}\overline{G}(0)=1+\rho\,\partial_{\varphi}\overline{f}(0).$
(2.32)

Noting that

$\overline{\Gamma}=\rho\,\overline{\gamma}:=-\rho\sup_{\varphi\in{\mathcal{U}}}\partial_{\varphi}\overline{f}(\varphi)\geq\rho\,\gamma=\Gamma,$

we see that the map $\overline{\mathscr{S}}$ satisfies Hypotheses 1–3 if
${\mathscr{S}}$ does.

######  Remark 2.25.

_Since the action of $\overline{\mathscr{S}}$ on each variable is independent
of the other one, in fact we have
$(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)=\overline{G}^{n}(\varphi)$.
Notwithstanding this, the notation introduced in (2.30) helps clarify the
forthcoming discussion. _

######  Remark 2.26.

_Noting that $\overline{f}(0)=0$ (see Remark 2.10), we see that
$\overline{W}(\psi)=0$ solves the equation
$\overline{\mathscr{S}}(\overline{W}(\psi),\psi)$
$=(\overline{W}(A_{0}\psi),A_{0}\psi)$, and hence $\overline{\mathscr{S}}$
admits the invariant manifold
$\overline{\mathcal{W}}=\\{(0,\psi),\psi\in\mathds{T}^{2}\\}$. _

In the two coming Subsections 2.4.4 and 2.4.5 we compare the evolution
generated by ${\mathscr{S}}$ with the evolution generated by
$\overline{\mathscr{S}}$. In the remainder of this subsection, we show that,
by adapting the analysis in Subsection 2.3.2 to the map
$\overline{\mathscr{S}}$, we are able to compare the trajectories of the
dynamics generated by $\overline{\mathscr{S}}$ with the solution of (2.29).

First, we proceed as in Theorem 2, and look for an invertible function

$\overline{\mathscr{H}}(\varphi,\psi)=(\overline{\mathcal{H}}(\varphi),\psi),\qquad\qquad\overline{\mathcal{H}}(\varphi)=\varphi+\varphi^{2}\overline{h}(\varphi),$
(2.33)

such that

$\overline{\mathscr{H}}{\circ}\overline{\mathscr{S}}=\overline{\mathscr{S}}_{0}{\circ}\overline{\mathscr{H}}\,,$
(2.34)

that is a function $\overline{\mathscr{H}}$ that conjugates
$\overline{\mathscr{S}}$ to $\overline{\mathscr{S}}_{0}$. Analogously to
(2.33), we also write

$\overline{\mathscr{H}}^{-1}(\eta,\psi)=(\eta+\eta^{2}\bar{l}(\eta),\psi).$
(2.35)

Then, in Subsection 7.6 we prove the following result.

######  Lemma 2.27.

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.15)
satisfying Hypotheses 1–3, and define the map $\overline{\mathscr{S}}$ as in
(2.30). There exist a closed interval ${\mathcal{U}}_{0}\subset\mathds{T}$ and
a function
$\overline{\mathscr{H}}\\!:{\mathcal{U}}\times\mathds{T}^{2}\to{\mathcal{U}}_{0}\times\mathds{T}^{2}$
such that (2.34) holds. Moreover, there exists a constant $C$ such that one
has
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\overline{h}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,3}\leq
C$ and
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\bar{l}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,3}\leq
C$.

######  Remark 2.28.

_Note that $\Omega_{0}\neq{\mathcal{U}}_{0}\times\mathds{T}^{2}$: in fact, one
has $\Omega_{0}={\mathscr{H}}(\Omega)$, while
${\mathcal{U}}_{0}=\overline{\mathcal{H}}({\mathcal{U}})$. _

We can now introduce the flow $\Phi$ generated by (2.29), that is the set of
the solutions of

$\left\\{\begin{aligned}
&\frac{d}{dt}\Phi_{t}(\varphi)=\overline{f}(\Phi_{t}(\varphi)),\\\
&\Phi_{0}(\varphi)=\varphi,\end{aligned}\right.$ (2.36)

when varying $\varphi\in{\mathcal{U}}$. Because of Hypotheses 1–3, all
trajectories $\Phi_{t}(\varphi)$ of the system (2.36), with
$\varphi\in{\mathcal{U}}$, move towards the origin at exponential rate as $t$
tend to infinity.

Observe that
$\Phi_{n\rho}=(\Phi_{\rho})^{n}=\Phi_{\rho}{\circ}\ldots{\circ}\Phi_{\rho}$
and that $\Phi_{\rho}(\varphi)-\overline{\mathscr{S}}(\varphi)$ is
$O(\rho^{2})$. The following lemma, proved in Subsection 7.7, show that the
trajectories generated by (2.30) and (2.36) remain close and, in fact, merge
asymptotically.

######  Lemma 2.29.

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3, and define the map $\overline{\mathscr{S}}$ as in
(2.30) and the flow $\Phi$ as in (2.36). For any
$\gamma^{\prime}\in(0,\gamma)$ there exists a positive constant $C$ such that
for all $\varphi\in{\mathcal{U}}$ and all $n>0$ one has

$|(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)-\Phi_{n\rho}(\varphi)|\leq
C\rho\,(1-\rho\gamma^{\prime})^{n}.$ (2.37)

######  Remark 2.30.

_With the scaling terminology introduced immediately before Subsection 2.4.1,
we can rewrite (2.37) as_

$|(\overline{\mathscr{S}}^{\lfloor
t/\rho\rfloor})_{\varphi}(\varphi,\psi)-\Phi_{t}(\varphi)|\leq
C\rho\,e^{-\gamma^{\prime}t}.$

_Observe that the decay rate in Lemma 2.29 cannot in general be equal to
$\gamma$ as one could naïvely expect. Indeed, the constant $C$ in Lemma 2.29,
as well as in the forthcoming Lemma 2.33 and Theorems 6 and 7, depends on
$\gamma^{\prime}$ and may diverge as $\gamma^{\prime}$ tends to $\gamma$. _

####  2.4.4 Oscillations and deviations in the scaling regime

In the next subsection we will show that also
$({\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)$ and $\Phi_{n\rho}(\varphi)$
remain close in the sense that the first and second moment, with respect to
$\psi$, of their difference are $O(\rho)$ uniformly in $n$. In this subsection
we present several preparatory results that, in our opinion, are also of
interest in their own. The proofs of these results form the main technical
part of the present work and are reported in Section 7. The main tools used in
these proofs are the decay of correlations estimates contained in Propositions
7.6 and 7.30.

We first show that even though the oscillations of ${\mathcal{W}}$ around
$\overline{\mathcal{W}}$ can be of order 1 in $\rho$, large oscillations are
rare, in the sense of the following result, proved in Subsection 7.3.

######  Theorem 4 (Oscillations of the invariant manifold).

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Let
${\mathcal{W}}=\\{(W(\psi),\psi):\psi\in\mathds{T}^{2}\\}$ be the invariant
manifold for the map ${\mathscr{S}}$ as in Theorem 1. Then, there is a
constant $C$ such that

$\left|\langle W\rangle\right|\leq C\rho,\qquad\qquad\langle W^{2}\rangle\leq
C\rho.$ (2.38)

######  Remark 2.31.

_From Theorem 4 and Chebyshev inequality we obtain that, for any $\delta>0$,_

$m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:|W(\psi)|>\delta\Bigr{\\}}\Bigr{)}\leq\frac{C\rho}{\delta^{2}}\,.$
(2.39)

_Therefore, the invariant manifold ${\mathcal{W}}$ for ${\mathscr{S}}$
converges in probability to the invariant manifold $\overline{\mathcal{W}}$ of
$\overline{\mathscr{S}}$. This also implies that most trajectories of
${\mathscr{S}}$ starting on ${\mathcal{W}}$ will spend most of their time very
close to $\varphi=0$ while only rarely venturing away._

Next we compare the linearized maps ${\mathscr{S}}_{0}$ and
$\overline{\mathscr{S}}_{0}$, defined in (2.22) and (2.32), respectively.
Observe that
$\overline{\mathscr{S}}_{0}^{n}(\varphi,\psi)=(\overline{\mu}^{n}\varphi,A_{0}^{n}\psi)$,
while
${\mathscr{S}}_{0}^{n}(\varphi,\psi)=(\kappa^{(n)}(\psi)\varphi,A_{0}^{n}\psi)$,
where

$\kappa^{(n)}(\psi):=\prod_{i=0}^{n-1}\kappa(A_{0}^{i}\psi)\,,$ (2.40)

with $\kappa(\psi)$ defined in (2.22).

######  Remark 2.32.

_Throughout the paper we use the convention that a product over an empty set
of indices is 1, while a sum over an empty set of indices is 0. In particular
this convention implies that $\kappa^{(0)}=1$ in (2.40). _

The following lemma, proved in Subsection 7.4, shows that the maps
${\mathscr{S}}_{0}^{n}$ and $\overline{\mathscr{S}}_{0}^{n}$ stay close to
each other uniformly in $n$; what makes the result not trivial is that the
function $\kappa$ in (2.40) is only in
${\mathfrak{B}}^{*}_{\alpha_{-},\alpha_{+}}(\mathds{T}^{2},\mathds{R})$, with
$\alpha_{-}=O(\rho)$ and $|W|^{-}_{\alpha_{-}}=O(1)$ in $\rho$.

######  Lemma 2.33.

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Let $\kappa^{(n)}(\psi)$ and $\overline{\mu}$ be
defined in (2.40) and in (2.32), respectively. Then for any
$\gamma^{\prime}\in(0,\gamma)$ there is a constant $C$ such that, for all
$n\in\mathds{N}$,

$\bigl{|}\langle\kappa^{(n)}-\overline{\mu}^{n}\rangle\bigr{|}\leq
C\rho\,(1-\rho\,\gamma^{\prime})^{n},\qquad\bigl{|}\langle\bigl{(}\kappa^{(n)}-\overline{\mu}^{n}\bigr{)}^{2}\rangle\bigr{|}\leq
C\rho\,(1-\rho\gamma^{\prime})^{2n}.$ (2.41)

Finally we want to estimate the deviation of ${\mathcal{H}}$ from
$\overline{\mathcal{H}}$. To this end, we show that $h(\varphi,\psi)$, defined
in (2.25), and $\overline{h}(\varphi,\psi)$, defined in (2.33), are close,
more precisely that both $h(\varphi,\psi)-\overline{h}(\varphi,\psi)$ and its
derivative are small – always in the sense that their first and second moments
are of order $\rho$. This is ensured by the following result, whose proof is
given in Subsection 7.8.

######  Theorem 5 (Deviations of the conjugation).

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Let $h$ and $\overline{h}$ be defined as in (2.25)
and in (2.33), respectively. Then, there is a constant $C$ such that for all
$\varphi\in{\mathcal{U}}$

$\displaystyle\left|\left\langle
h(\varphi,\cdot)-\overline{h}(\varphi)\right\rangle\right|$ $\displaystyle\leq
C\rho,$
$\displaystyle\left\langle(h(\varphi,\cdot)-\overline{h}(\varphi))^{2}\right\rangle$
$\displaystyle\leq C\rho,$ (2.42a)
$\displaystyle\left|\left\langle\partial_{\varphi}h(\varphi,\cdot)-\partial_{\varphi}\overline{h}(\varphi)\right\rangle\right|$
$\displaystyle\leq C\rho,$
$\displaystyle\left\langle(\partial_{\varphi}h(\varphi,\cdot)-\partial_{\varphi}\overline{h}(\varphi))^{2}\right\rangle$
$\displaystyle\leq C\rho.$ (2.42b)

Bounds analogous to (2.42) hold also for the deviations of the function $l$,
defined in (2.25), from $\overline{l}$, defined in (2.35); we refer to
Subsection 7.8 – and to Proposition 7.57 in particular – for a precise
statement, which requires introducing a suitable extension of the map
${\mathscr{S}}$ along the lines considered in Remark 2.12.

######  Remark 2.34.

_It is in order to prove the bounds in Theorem 5 – and in the forthcoming
Proposition 7.57 – that we need $F$ to be in
${\mathscr{B}}_{\alpha_{0},6}(\Omega,\mathds{R})$; see also Remark 7.51 in
Subsection 7.8. _

####  2.4.5 Summing up: convergence in square mean and in probability

We can now complete the comparison of the evolution generated by
${\mathscr{S}}$ with that generated by $\overline{\mathscr{S}}$. From (2.23)
and (2.34) we get

$\big{(}{\mathscr{S}}^{n}\big{)}_{\varphi}(\varphi,\psi)-(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)={\mathcal{H}}^{-1}\bigl{(}{\mathscr{S}}_{0}^{n}({\mathcal{H}}(\varphi,\psi),\psi)\bigr{)}-\overline{\mathcal{H}}^{-1}\bigl{(}\overline{\mathscr{S}}_{0}^{n}(\overline{\mathcal{H}}(\varphi),\psi)\bigr{)}.$
(2.43)

Combining the estimates in Lemma 2.33 with the bounds in Theorem 5 and the
analogous bounds for the inverse conjugation in Proposition 7.57, in
Subsection 7.10 we prove our main result on the relation between the dynamics
generated by ${\mathscr{S}}$ and the averaged dynamics generated by
$\overline{\mathscr{S}}$.

######  Theorem 6 (Convergence in square mean).

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Then for any $\gamma^{\prime}\in(0,\gamma)$ there
exists a constant $C$ such that, for all $n\in\mathds{N}$ and all
$\varphi\in{\mathcal{U}}$,

$\displaystyle\left|\left\langle({\mathscr{S}}^{n})_{\varphi}(\varphi,\cdot)-\bigl{(}(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi,\cdot)+W(A_{0}^{n}\cdot)\bigr{)}\right\rangle\right|$
$\displaystyle\leq C\rho\,(1-\rho\,\gamma^{\prime})^{n},$ (2.44a)
$\displaystyle\Big{\langle}{\Bigl{(}({\mathscr{S}}^{n})_{\varphi}(\varphi,\cdot)-\bigl{(}(\overline{\mathscr{S}}^{n})_{\varphi}(\varphi,\cdot)+W(A_{0}^{n}\cdot)\bigr{)}\Bigr{)}^{\\!2}}\Big{\rangle}$
$\displaystyle\leq C\rho\,(1-\rho\,\gamma^{\prime})^{2n}.$ (2.44b)

######  Remark 2.35.

_The bound ( 2.44b) for the second moment of the fluctuations and the Cauchy-
Schwartz inequality trivially would imply a weaker bound than (2.44a). On the
contrary, proving that also the first moment of the fluctuations is of order
$\rho$ requires a substantially greater amount of work. A similar comment
holds for the results in Theorem 4 and, in fact, it applies 5 to Lemma 2.33
above as well._

The following result is an immediate consequence of Theorems 4 and 6, and
Lemma 2.29.

######  Corollary 2.36.

Under the hypotheses of Theorem 6, there exists a constant $C$ such that, for
all $n\in\mathds{N}$ and all $\varphi\in{\mathcal{U}}$, one has

$\Bigl{|}\left\langle({\mathscr{S}}^{n})_{\varphi}(\varphi,\cdot)-\Phi_{n\rho}(\varphi)\right\rangle\Bigr{|}\leq
C\rho,\qquad\qquad\Big{\langle}{\Bigl{(}({\mathscr{S}}^{n})_{\varphi}(\varphi,\cdot)-\Phi_{n\rho}(\varphi)\Bigr{)}^{2}}\Big{\rangle}\leq
C\rho,$

with the flow $\Phi$ defined as in (2.36).

Similarly to Remark 2.31, from Theorem 6, for fixed $\varphi$ and $n$, we
obtain that, for any $\gamma^{\prime}\in(0,\gamma)$ and for any $\delta>0$,

$m_{0}\Bigl{(}\Bigl{\\{}\psi\in\mathds{T}^{2}:(1-\rho\,\gamma^{\prime})^{-n}\bigl{|}({\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)-\bigl{(}\Phi_{n\rho}(\varphi)+W(A_{0}^{n}\psi)\bigr{)}\bigr{|}>\delta\Bigr{\\}}\Bigr{)}\leq\frac{C\rho}{\delta^{2}}.$
(2.45)

Note that the set of angles $\psi$ considered in (2.45) depends on $n$, even
though its measure is bounded independently of the value of $n$. The following
theorem shows that, for most most values of $\psi$, the difference between
$({\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)$ and
$\Phi_{n\rho}(\varphi)+W(A_{0}^{n}\psi)$ is small, exponentially in $n$.

######  Theorem 7 (Convergence in probability, I).

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Let the flow $\Phi$ be defined as in (2.36). Then
for any $\gamma^{\prime}\in(0,\gamma)$ there exists a constant $C$ such that,
for all $\varphi\in{\mathcal{U}}$,

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

The proof of Theorem 7 is in Subsection 8.1.

####  2.4.6 Aftermath: continuous time

We now give a more probabilistic description of the results in Theorem 7 on
the relations between ${\mathscr{S}}^{\lfloor t/\rho\rfloor}$ and the
solutions of (2.29) and express them in terms of the rescaled time $t$. To
this aim, for every $\varphi\in\mathds{T}$ and every $t\in\mathds{R}^{+}$, we
consider the random variable $X_{t}\\!:\mathds{T}^{2}\to\mathds{T}$ defined as

$X_{t}(\psi):=({\mathscr{S}}^{\lfloor
t/\rho\rfloor})_{\varphi}(\varphi,\psi)+(t/\rho-\lfloor
t/\rho\rfloor)\left(({\mathscr{S}}^{\lfloor
t/\rho\rfloor+1})_{\varphi}(\varphi,\psi)-({\mathscr{S}}^{\lfloor
t/\rho\rfloor})_{\varphi}(\varphi,\psi)\right).$ (2.46)

Observe that for every $\psi$, $X_{t}(\psi)$ is a continuous function of $t$
so that $X_{t}$ can be seen as a stochastic process with trajectories in
$C^{0}(\mathds{R}^{+},\mathds{T})$. Similarly we consider the process
$\widetilde{X}_{t}$ with trajectories in $C^{0}(\mathds{R}^{+},\mathds{T})$
defined as

$\widetilde{X}_{t}(\psi)=X_{t}(\psi)-W(A_{0}^{\lfloor
t/\rho\rfloor}\psi)-(t/\rho-\lfloor t/\rho\rfloor)\left(W(A_{0}^{\lfloor
t/\rho\rfloor+1}\psi)-W(A_{0}^{\lfloor t/\rho\rfloor}\psi)\right).$ (2.47)

We want to compare the stochastic processes $X_{t}$ and $\widetilde{X}_{t}$
with the flow $\Phi_{t}$ defined in (2.36), seen as a stochastic process on
$C^{0}(\mathds{R}^{+},\mathds{T})$.

To compare $\widetilde{X}_{t}$ with $\Phi_{t}$, for $x\in
C^{0}(\mathds{R}^{+},\mathds{T})$, we consider the norm

$\|x\|_{\rm exp}:=\sup_{t\geq 0}\,e^{\xi t}|x(t)|\,,$ (2.48)

with a suitable $\xi\geq 0$. Combining Lemma 2.29 and Theorem 7 provides the
following result (see Subsection 8.2 for the proof).

######  Theorem 8 (Convergence in probability, II).

Consider the dynamical system described by the map ${\mathscr{S}}$ in (2.26)
satisfying Hypotheses 1–3. Let $\widetilde{X}_{t}$ and $\Phi_{t}$ the
stochastic process (2.47) and the flow defined in (2.36), respectively. Then
one has

$\lim_{\rho\to 0^{+}}\widetilde{X}_{t}=\Phi_{t},$

where the limit is taken in probability in the topology on
$C^{0}(\mathds{R}^{+},\mathds{T})$ generated by the norm (2.48) for any
$\xi\in[0,\gamma)$.

We close our results with an immediate consequence of Theorem 8.

######  Corollary 2.37.

Under the hypotheses of Theorem 8, let $X_{t}$ be the stochastic process
defined in (2.46). Then

$\lim_{\rho\to 0^{+}}X_{t}=\Phi_{t},$

where the limit is taken in probability in the topology of the uniform
convergence in $C^{0}(\mathds{R}^{+},\mathds{T})$.

###  2.5 Content of the paper and strategy of the proof

The rest of the paper is mainly devoted to the proof of the results stated
above, with the exception of Sections 3 to 5, where the relation with the
existing literature is examined.

In Section 3 we report on previous results on similar models, including
results by one of the authors where more restrictive hypotheses were assumed,
and results for systems where one-dimensional expanding maps are considered
instead of Anosov automorphisms, while in Section 4 we discuss a few open
problems and possible extensions of our work, also in relation with the kind
of problems which are mainly investigated in the literature. Next, in Section
5 we briefly review a few fields of possible application in problems of
physical interest, such as the combination of slow and fast motions in
describing the effects of weather on climate and the derivation of the heat
equation.

The remaining, more technical Sections 6 to 8, which represent the core of the
paper, are organized as follows.

Section 6 contains the proofs of the Theorems 1 to 3, together with the
derivation of the properties of both the invariant manifold and the
conjugation that will be needed for dealing with the scaling regime. In
particular the existence of the invariant manifold is formulated as a fixed-
point problem in a suitable Banach space (Theorem 1). Thereafter, the
conjugation is shown to admit a series representation which is studied and
proved to converge to a function which satisfies the properties stated in
Theorem 2. This result then implies the existence of the inverse conjugation
as well (Corollary 2.18), which satisfies similar properties, and the mixing
properties of the system while evolving toward the invariant manifold (Theorem
3). Technically it turns out to be useful to use the coordinates
$(\theta,\psi)$, with $\theta:=\varphi-W(\psi)$ and $W(\psi)$ as in Subsection
2.3.1, in terms of which the dynamics is described by the map

${\mathscr{S}}_{1}(\theta,\psi)=(\theta+F(\theta+W(\psi),\psi)-F(W(\psi),\psi),A_{0}\psi),$

that we call the _translated map_ , and the attracting invariant manifold is
the flat torus $\mathds{T}^{2}$.

In Section 7 we discuss the averaging problem in the scaling regime: setting
$F(\varphi,\theta)=\rho f(\varphi,\theta)$, the dynamics of
${\mathscr{S}}^{k}(\varphi,\psi)$ is studied for $k=\lfloor t/\rho\rfloor$,
with $t$ fixed and $\rho\to 0^{+}$. After deriving, as a preliminary step, the
deviation laws for the invariant manifold (Theorem 4) and the conjugation
(Theorem 5), we provide the proof of Theorem 6 on the deviations of the
dynamics with respect to that of the averaged system. The analysis is based on
delicate correlation inequalities, which make use of the map being weakly
dissipative and the ensuing fact that any trajectory after a while comes close
enough to the attracting invariant manifold. A main issue – and a major source
of technical intricacies – is that the correlations inequalities are related
to the regularity of the involved functions. In general, we have to deal with
averages of the form $\langle g_{+}g_{-}{\circ}{\mathscr{S}}_{1}^{n}\rangle$,
with $g_{-}\in{\mathscr{B}}_{\alpha}^{-}(\Omega,\mathds{R})$, for which we can
expect decay properties analogous to those of Proposition 2.3. However, the
invariant manifold and, hence, the map ${\mathscr{S}}_{1}$ are only
$\alpha$-Hölder continuous, with $\alpha=O(\rho)$, so that a naiv̈e
generalization of Proposition 2.3 would provide unavailing bounds. If we
expand the average $\langle g_{+}g_{-}{\circ}{\mathscr{S}}_{1}^{n}\rangle$ to
second order in $W$, only the first order contributions require a careful
analysis, since the leading terms are regular and the second order terms can
be dealt with by using the bounds on the variance provided by Theorem 4. Thus,
in order to use the bounds on the average in (2.38) we need to study in detail
the coefficients of the linear terms in the expansion in $W$. We achieve this
by isolating the contributions which do not depend on the dynamics on the
torus, that is on the chaotic variable $\psi$, and showing that the remaining
contributions admit better dimensional bounds. In practice, to implement the
scheme outlined above, we introduce a regularized version ${\mathscr{S}}_{2}$
of the translated map, that we call the _auxiliary map_ , for which we can
apply the correlation inequalities. Then we compare the translated map to the
auxiliary map through a series of technical lemmas which aim to extract and
study the linear dependence on the function $W$; this will be treated in
Subsection 7.8.4 and in Appendix D. A major issue, from a technical point of
view, is that we want that the map ${\mathscr{S}}_{2}$ regularizes
${\mathscr{S}}_{1}$ and, at the same time, still satisfies Hypotheses 1 to 3:
achieving both goals leads to contributions which, albeit depending linearly
on $W$, so that they be dealt with as outlined before, unfortunately contain
an extra factor $\rho^{-1}$. However such contributions involve sums of terms
in which there appear differences of functions $W$ and the sums can be
rearranged in such a way that the difference is shifted to more regular
functions: this allows us to regain a further factor $\rho$ so as to
compensate the factor $\rho^{-1}$. To implement the idea described above we
need to perform iterated expansions which make the analysis rather intricate:
in fact, Subsection 7.8.4 and Appendix D constitute the most technical part of
the paper.

Finally, in Section 8, we prove Theorems 7 and 8 on the asymptotic behavior of
the stochastic process associated to the dynamics. Again, the crucial issue is
that in average the deviations are small, and hence the evolution of the
system is essentially determined by the averaged map.

Appendix A contains the results on the decay of correlations for the evolution
generated by $A_{0}$ on $\mathds{T}^{2}$. Such results form the main toolkit
we use in Section 7 to obtain more general correlation inequalities.
Appendices B, C and – as said above – mainly D contain mostly the proofs of
the more technical results presented in Section 7.

##  3 Relation with previous works

###  3.1 Synchronization under stronger assumptions

In the scaling regime, dynamical systems described by (2.26) are a
generalisation of the systems studied in ref. [30], i.e. continuous systems
defined on $\mathds{T}\times\mathds{T}^{2}$ of the form

$\begin{cases}\dot{\varphi}=1+\varepsilon g(\varphi,\psi,t),\\\
\dot{\psi}=\delta(t)\,\log A_{0}\,\varphi,\end{cases}$ (3.1)

where $\delta(t)$ is the $2\pi$-periodic delta function so defined that its
integral from 0 to $t$ equals 1 for any $t>0$. By integrating the equations
(3.1) up to time $2\pi$ and using that $\varphi$ is defined mod $2\pi$, we
find the Poincaré map

$\begin{cases}\displaystyle{\varphi(2\pi)=\varphi(0)+\varepsilon\int_{0}^{2\pi}{\rm
d}t\,g\Bigl{(}\varphi(0)+t+\varepsilon\int_{0}^{t}{\rm
d}s\,g(\varphi(s),A_{0}\psi,s),A_{0}\psi,t\Bigr{)}},\\\
\psi(2\pi)=A_{0}\,\psi(0).\raisebox{5.69046pt}{\phantom{\Big{)}}}\end{cases}$
(3.2)

If we set $(\varphi,\psi):=(\varphi(0),\psi(0))$ and
${\mathscr{S}}_{c}(\varphi,\psi):=(\varphi(2\pi),\psi(2\pi))$, we see that
${\mathscr{S}}_{c}$ is of the form (2.15), that is

${\mathscr{S}}_{c}(\varphi,\psi)=(\varphi+F_{c}(\varphi,\psi),A_{0}\,\psi),$

with
$F_{c}(\varphi,\psi)=\varepsilon\,F_{1}(\varphi,\psi)+\varepsilon^{2}F_{2}(\varphi,\psi,\varepsilon)$,
where

$\displaystyle F_{1}(\varphi,\psi)$ $\displaystyle\\!\\!\\!:=\\!\\!\\!$
$\displaystyle\int_{0}^{2\pi}{\rm d}t\,g(\varphi+t,A_{0}\psi,t),$
$\displaystyle F_{2}(\varphi,\psi,\varepsilon)$
$\displaystyle\\!\\!\\!:=\\!\\!\\!$ $\displaystyle\int_{0}^{2\pi}{\rm
d}t\,\partial_{\varphi}g\bigl{(}\varphi^{*}(t),A_{0}\psi,t\bigr{)}\int_{0}^{t}{\rm
d}s\,g(\varphi(s),A_{0}\psi,s),$

for a suitable function $\varphi^{*}(t)$. The existence of an invariant
manifold for the system (3.1) is proved in ref. [30] under the hypothesis that
there exists $\overline{\varphi}\in\mathds{T}$ such that one has

$F_{1}(\overline{\varphi},\psi)=0,\qquad\qquad\partial_{\varphi}F_{1}(\overline{\varphi},\psi)<0,$
(3.3)

for all $\psi\in\mathds{T}^{2}$. Both $F_{1}$ and $F_{2}$ are smooth in
$\varphi$ and $\psi$, and it is easy to see that ${\mathscr{S}}_{c}$ satisfies
Hypotheses 1–3 with $\Gamma=O(\varepsilon)$. In particular the assumptions
(3.3) imply that $F_{c}(\bar{\varphi},\psi)=O(\varepsilon^{2})$: this implies
that the invariant manifold is such that $W_{c}(\psi)=O(\varepsilon)$, that is
the oscillations of the invariant manifold are small not only in average.
Therefore, systems of the form (2.15) extend the class of systems considered
in ref. [30]; in fact Theorem 1 provides a positive answer to a question
raised in ref. [30], by showing that the assumptions on the functions
$\gamma_{0}(\varphi,\psi):=F_{1}(\varphi,\psi)$ and
$\gamma_{1}(\varphi,\psi):=\partial_{\varphi}F_{1}(\varphi,\psi)$ can be
weakened as suggested therein.

###  3.2 Expanding maps

Our work is also strongly related to the analyses in refs. [22, 23, 15]. In
these works a family of one-parameter maps ${\mathscr{T}}$ on
$\mathds{T}\times\mathds{T}$ is considered, given by

${\mathscr{T}}(\varphi,\omega):=(\varphi+\rho
f(\varphi,\omega),h(\varphi,\omega)),$ (3.4)

where $h(\varphi,\cdot)$ is an expanding circle map for every fixed $\varphi$.
Systems of the form (3.4), notwithstanding the fact that they are not time-
reversible and have no Hamiltonian structure, have been extensively
investigated since a full understanding of their behaviour may be considered
as an important step toward the study of more realistic models.

Although the hypotheses on $h$ and $f$ in refs. [22, 23] are much weaker then
ours, it is interesting to compare their results with ours. For this purpose,
we consider the dynamical system ${\mathscr{S}}_{b}$ of the form (2.15) but
with the baker transformation in place of the automorphism $A_{0}$. That is,
we consider the dynamical system on $\mathds{T}\times\mathds{T}^{2}$ given by

${\mathscr{S}}_{b}(\varphi,\psi):=(\varphi+\rho
f(\varphi,\psi),B(\psi)),\qquad B(\psi):=\left(2\psi_{1}\;{\rm
mod}\;2\pi,\frac{\psi_{2}+\lfloor 2\psi_{1}\rfloor}{2}\right),$

where $\psi=(\psi_{1},\psi_{2})$ and $\lfloor\cdot\rfloor$ denotes the lower
integer part, and assume that ${\mathscr{S}}_{b}$ satisfies Hypotheses 1–3. If
$f(\varphi,\psi)$ does not depend on $\psi_{2}$ then the system
${\mathscr{S}}_{b}$ can be seen as an extension of the system
${\mathscr{T}}_{b}(\varphi,\psi_{1}):=(\varphi+\rho
f(\varphi,\psi),2\psi_{1}\;{\rm mod}\;2\pi)$ that is part of the family of
systems studied in refs. [22, 23]. On the other hand, we expect that most of
the results in the present paper apply with minor modifications to
${\mathscr{S}}_{b}$; that is, there exists an invariant manifold
$\mathcal{W}_{b}:=\\{(\psi,W_{b}(\psi)):\psi\in\mathds{T}^{2}\\}$, with
$W_{b}\\!:\mathds{T}^{2}\to\mathds{T}$ Hölder continuous such that
${\mathscr{S}}_{b}(W_{b}(\psi),\psi)=W_{b}(B(\psi))$, and $\mathcal{W}_{b}$ is
a global attractor for ${\mathscr{S}}_{b}$ and hence ${\mathscr{S}}_{b}$
admits a unique physical measure given by

$\nu_{b}(\mathcal{O})=\int\mathcal{O}(W_{b}(\psi),\psi)\,m_{0}({\rm
d}\psi)=\int\mathcal{O}(\varphi,\psi)\,\delta(\varphi-
W_{b}(\psi))\,\nu_{0}({\rm d}\varphi{\rm d}\,\psi)\,.$

In particular, when $f(\varphi,\psi)$ does not depend on $\psi_{2}$, if we
take $\mathcal{O}$ independent of $\psi_{2}$ as well, we can write

$\nu_{b}(\mathcal{O})=\int\mathcal{O}(\varphi,\psi_{1})\,\bar{\nu}_{b}({\rm
d}\varphi\,{\rm d}\psi_{1}),$

where $\bar{\nu}_{b}$ is the projection of $\nu_{b}$ along the
$\psi_{2}$-direction and it is the unique physical measure for
${\mathscr{T}}_{b}$. In ref. [15] the authors show that, under very general
conditions, $\bar{\nu}_{b}$ is absolutely continuous w.r.t. the Lebesgue
measure $\bar{\nu}_{0}={\rm d}\psi_{1}\,{\rm d}\varphi$, that is $\nu_{b}({\rm
d}\varphi\,{\rm d}\psi_{1})=n_{b}(\varphi,\psi_{1}){\rm d}\varphi\,{\rm
d}\psi_{1}$, with

$n_{b}(\varphi,\psi_{1})=\int\delta(\varphi-W_{b}(\psi))\,{\rm d}\psi_{2}$

a well-defined integrable function. We think that, in our case, this follows
from the Hölder continuity of $W_{b}(\psi)$ and the fact that it varies
rapidly in the $\psi_{2}$ direction; a formal derivation of this property is
beyond the scope of the present paper.

Moreover, an adaptation of Corollary 2.24 to the context under consideration
should imply that, for every observable ${\mathcal{O}}_{1}$ and
${\mathcal{O}}_{2}$ on $\mathds{T}\times\mathds{T}$ that are $C^{1}$ in
$\varphi$ and Hölder continuous in $\psi_{1}$, we have

$\left|\int{\mathcal{O}}_{1}(\varphi,\psi_{1}){\mathcal{O}}_{2}({\mathscr{T}}_{b}^{n}(\varphi,\psi_{1}))\,{\rm
d}\varphi\,{\rm d}\psi_{1}-\int{\mathcal{O}}_{1}(\varphi,\psi_{1})\,{\rm
d}\varphi\,{\rm
d}\psi_{1}\int{\mathcal{O}}_{2}(\varphi,\psi_{1})\,\bar{\nu}_{0}({\rm
d}\varphi\,{\rm d}\psi_{1})\right|\leq Ce^{-c\rho n},$

for suitable constants $C$ and $c$, not depending on $\rho$. Similarly,
calling

$\overline{X}_{t}(\psi_{1}):=({\mathscr{T}}_{b}^{\lfloor
t/\rho\rfloor})_{\varphi}(\varphi_{0},\psi_{1})+(t/\rho-\lfloor
t/\rho\rfloor)\left(({\mathscr{T}}_{b}^{\lfloor
t/\rho\rfloor+1})_{\varphi}(\varphi_{0},\psi_{1})-({\mathscr{T}}_{b}^{\lfloor
t/\rho\rfloor})_{\varphi}(\varphi_{0},\psi_{1})\right),$

a result analogous to Corollary 2.37 should imply that $\overline{X}_{t}$
converges in probability to $\Phi_{t}$ in the topology of uniform convergence
in $C^{0}(\mathds{R}^{+},\mathds{T})$. Thus, the presence of uniform
contraction near $\mathcal{W}_{b}$ would allow to control the dynamics for all
positive times and hence to obtain stronger results with respect to the models
considered in ref. [23] (see in particular ref. [23, Section 3.1]).

##  4 Extensions and generalizations

As said in the introduction, we consider this work as a first step to fix
techniques and strategies to be applied to more general system and/or more
refined questions. In this section we present some of these questions and
briefly discuss possible strategies to follow in order to solve them by
applying the results of this paper.

###  4.1 More general perturbations: fully coupled systems

The systems considered in this work are usually called _skew products_ since
the fast variable does not depend on the slow variable. In this subsection we
present a path to generalize our results to the case of a fully coupled
system.

####  4.1.1 More general Anosov diffeomorphisms

As a first step we can consider systems of the form

${\mathscr{S}}(\varphi,\psi)=(\varphi+F_{\varphi}(\varphi,\psi),{\mathcal{A}}(\psi)),\qquad{\mathcal{A}}(\psi):=A_{0}\psi+F_{\psi}(\psi),$
(4.1)

with $F_{\psi}\\!:\mathds{T}^{2}\to\mathds{T}^{2}$ such that the map
$\psi\mapsto{\mathcal{A}}(\psi)$ describes an Anosov diffeomorphism on
$\mathds{T}^{2}$, and
$F_{\varphi}(\varphi,\psi)\\!:\mathds{T}\times\mathds{T}^{2}\to\mathds{T}$
such that ${\mathscr{S}}$ still satisfies Hypotheses 1–3. However, any Anosov
diffeomorphism of the form in (4.1) is conjugated with its linear part $A_{0}$
(see also refs. [10, 28, 45] for a more general context). Thus, there exists a
Hölder continuous map
$\widetilde{\mathcal{H}}_{\psi}\\!:\mathds{T}^{2}\to\mathds{T}^{2}$ such that

$\widetilde{\mathcal{H}}_{\psi}{\circ}{\mathcal{A}}=A_{0}{\circ}\widetilde{\mathcal{H}}_{\psi}\,.$

If we decompose $F_{\psi}(\psi)=F_{\psi,+}(\psi)v_{+}+F_{\psi,-}(\psi)v_{-}$,
where $v_{+}$ and $v_{-}$ are the eigenvectors corresponding to the
eigenvalues $\lambda>1$ and $\lambda^{-1}$ of $A_{0}$ (see Subsection 2.1),
and write
$\widetilde{\mathcal{H}}_{\psi}(\psi)=\psi+\widetilde{h}_{\psi}(\psi)$, with
$\widetilde{h}_{\psi}(\psi)=\widetilde{h}_{\psi,+}(\psi)v_{+}+\widetilde{h}_{\psi,-}(\psi)v_{-}$,
we find [29]

$\widetilde{h}_{\psi,+}(\psi)=\sum_{n=0}^{\infty}F_{\psi,+}({\mathcal{A}}^{n}(\psi))\lambda^{-(n+1)},\qquad\widetilde{h}_{\psi,-}(\psi)=-\sum_{n=1}^{\infty}F_{\psi,+}({\mathcal{A}}^{-n}(\psi))\lambda^{-(n+1)},$
(4.2)

so that, setting
$\widetilde{\mathscr{H}}(\varphi,\psi):=(\varphi,\widetilde{\mathcal{H}}_{\psi}(\psi))$,
we may use $\widetilde{\mathscr{H}}$ to conjugate ${\mathscr{S}}$ in (4.1)
with

$\widetilde{\mathscr{S}}(\varphi,\psi)=(\varphi+\widetilde{F}(\varphi,\psi),A_{0}\psi),$
(4.3)

with
$\widetilde{F}(\varphi,\widetilde{\mathcal{H}}_{\psi}(\psi))=F_{\varphi}(\varphi,\psi)$.
Clearly $\widetilde{\mathscr{S}}$ is of the form (2.15) and satisfies
Hypotheses 1–3.

In this situation it is still be natural to chose the initial $\psi$
distributed according to the Lebesgue measure $m_{0}$ on $\mathds{T}^{2}$.
This is essentially equivalent to considering the SRB measure
$m_{\mathcal{A}}$ associated with ${\mathcal{A}}$, since
${\mathcal{A}}^{*}m_{0}$ converges exponentially fast to $m_{\mathcal{A}}$
[29]. Thus, to apply the results in Section 2 to ${\mathscr{S}}$ in (4.1), we
need decay of correlations estimates like those in Proposition 2.3 but with
$\widetilde{\mathcal{H}}^{*}m_{\mathcal{A}}$ in place of $m_{0}$. Observe that
$\widetilde{\mathcal{H}}$ is Hölder continuous and
$\widetilde{\mathcal{H}}^{*}m_{\mathcal{A}}$ is invariant under the action of
$A_{0}$; thus $m_{\mathcal{A}}$ can be represented as a Gibbs state on the
same subshift of finite type used for $A_{0}$ in Appendix A.1. We can now
extend the proof Proposition 2.3 using the properties of the potential that
generates such a Gibbs state as discussed in ref. [29].

####  4.1.2 Bidirectional perturbations

We can now look at systems with bidirectional perturbations of the form

${\mathscr{S}}(\varphi,\psi)=(\varphi+F_{\varphi}(\varphi,\psi),\mathcal{A}(\psi;\varphi)),\qquad\mathcal{A}(\psi;\varphi):=A_{0}\psi+F_{\psi}(\varphi,\psi)),$
(4.4)

with $F_{\varphi}(\varphi,\psi)$ satisfying once more Hypotheses 1–3 and
$F_{\psi}(\varphi,\psi)$ such that the dynamical system on $\mathds{T}^{2}$
generated by the map $\psi\mapsto\mathcal{A}(\psi;\varphi)$ is an Anosov
diffeomorphism for every fixed $\varphi\in[\phi_{m},\phi_{M}]$. Here we
briefly sketch how the analysis of the present paper could be adapted to cover
such a more general situation

Existence of an invariant manifold for the system (4.4) is proven in ref. [17]
in the perturbative regime under the conditions (3.3). Note that, even in the
simpler case considered in ref. [17], the invariant manifold must be looked
for in the form ${\mathcal{W}}=\\{(W(\psi),H(\psi)):\psi\in\mathds{T}^{2}\\}$,
with $W$ and $H$ such that
${\mathscr{S}}(W(\psi),H(\psi))=(W(A_{0}\psi),H(A_{0}\psi))$, since the
dynamics of the chaotic variable is no longer trivial.

More generally, we can proceed as in Subsection 4.1.1 , and look for a
conjugation
$\widetilde{\mathscr{H}}(\varphi,\psi)=(\varphi,\widetilde{\mathcal{H}}_{\psi}(\varphi,\psi))$
such that

$\widetilde{\mathscr{H}}{\circ}{\mathscr{S}}=\widetilde{\mathscr{S}}{\circ}\widetilde{\mathscr{H}},$

with $\widetilde{\mathscr{S}}$ of the form (4.3). This means that

$\displaystyle\widetilde{\mathcal{H}}_{\psi}(\varphi+F_{\varphi}(\varphi,\psi),{\mathcal{A}}(\psi;\varphi))$
$\displaystyle=A_{0}\widetilde{\mathcal{H}}_{\psi}(\varphi,\psi),$ (4.5)
$\displaystyle\widetilde{F}(\varphi,\widetilde{\mathcal{H}}_{\psi}(\varphi,\psi))=F(\varphi,\psi).$

It is possible to write a formal solution to (4.5) along the lines of (4.2)
(see also [11] for a similar argument). Applying the conjugation
$\widetilde{\mathscr{H}}$ to the system (4.4) allows us to reduce it to a
system of the form of (2.15). To apply the results of the present paper one
then needs to prove that the resulting system has the geometric and regularity
properties needed to satisfy Hypotheses 1–3. The above construction, assuming
it is successful, allows to extend the results on synchronization to the
systems as in (4.4).

Then, as the next step, one must show that the averaging principle proved for
the system (2.15) implies, thanks to the existence of
$\widetilde{\mathscr{H}}$, an averaging principle for (4.4). More precisely,
writing $F_{\varphi}(\varphi,\psi)=\rho\,f_{\varphi}(\varphi,\psi)$ and
starting the evolution at $\varphi_{0}$, with $\psi$ distributed according to
$m_{0}$, the long time evolution generated by (4.4) for small $\rho$ should be
described by the flow $\widetilde{\Phi}$ defined by

$\left\\{\begin{aligned} &\frac{{\rm d}}{{\rm
d}t}\widetilde{\Phi}_{t}(\varphi_{0})=\widetilde{f}(\widetilde{\Phi}_{t}(\varphi_{0})),\\\
&\widetilde{\Phi}_{0}(\varphi)=\varphi_{0},\end{aligned}\right.$ (4.6)

with

$\widetilde{f}(\varphi):=\int_{\mathds{T}^{2}}f(\varphi,\psi)\,m_{\varphi}({\rm
d}\psi),$

where the new measure $m_{\varphi}$ can be computed from
$\widetilde{\mathcal{H}}_{\psi}$ in (4.2) and, as heuristic arguments suggest,
it is expected to be the SRB measure of the Anosov diffeomorphism
${\mathcal{A}}(\cdot;\varphi)$.

####  4.1.3 Non-dissipative perturbations

In our present work, and in the generalizations discussed above, the uniform
dissipation around $\varphi=0$ of the map $F$ plays an important role. A third
possible – and harder – generalization to investigate, already in the skew
product case, is obtained by weakening the hypotheses on the dissipative
nature of the map ${\mathscr{S}}$. Our Hypothesis 3 requires the map to be
strictly contracting in $\Lambda$. In refs. [22, 23], where expanding maps are
considered instead of Anosov automorphisms, a more general interaction is
investigated, since the rate of contraction of the neutral variable is assumed
to be non-zero only in average.

In fact, a very interesting case, from a physical point of view, is the
conservative one (see also the comments in Subsection 5.3), where one assume
that $\langle f(\cdot,\varphi)\rangle$ vanishes for all $\varphi$. In such a
situation one expects the correct scaling to be $k=\lfloor t/\rho^{2}\rfloor$
and to lead in the limit to a stochastic differential equation [18, 27],
instead of an ordinary differential equation as in the dissipative case.

###  4.2 Central Limit Theorem

It would be also of interest to find a more detailed description of the
fluctuations of the process $X_{t}$ in (2.46) around the flow $\Phi_{t}$ in
(2.36). Comparing with available results in the literature [36, 26, 27, 20,
21, 22, 23], for systems of the form (3.4), we expect the stochastic process

$\Delta_{t}:=\frac{1}{\sqrt{\rho}}(X_{t}-\Phi_{t})$

to converge in distribution, as $\rho\to 0^{+}$, to the solution of the
stochastic differential equation

$\begin{cases}\displaystyle{{\rm
d}\Delta_{t}=\partial_{\varphi}\overline{f}(\Phi_{t}(\varphi_{0}))\Delta_{t}{\rm
d}t+\sigma(\Phi_{t}(\varphi_{0}))\,{\rm d}B(t)},\phantom{\sum_{n}}&\\\
\Delta_{0}=0,\phantom{\sum}&\end{cases}$ (4.7)

where $B(t)$ is a standard Brownian motion and

$(\sigma(\varphi))^{2}:=\biggl{\langle}\sum_{n=-\infty}^{\infty}f(\varphi,\cdot)f(\varphi,A_{0}^{n}\cdot)\biggr{\rangle}\,.$

Since in the case of conservative interactions the scaling limit is expected
to lead to a stochastic differential equation (see the end comments in
Subsection 4.1.3), studying how an equation like (4.7) emerges from (2.15) in
the scaling regime can be seen as a precursory step before dealing with the
more demanding scaling regime needed to study conservative systems.

The fact that the stochastic process $\Delta_{t}$ converges to the solution of
equation (4.7) implies that $W/\sqrt{\rho}$, with $W$ (see Subsection 2.3.1)
seen as a random variable on $\mathds{T}^{2}$, converges in distribution to a
normal random variable with mean 0 and standard deviation $\sigma(0)$, that is

$\lim_{\rho\to
0^{+}}m_{0}\left(\\{\psi\in\mathds{T}^{2}:W(\psi)\leq\sqrt{\rho}z\\}\right)=\frac{1}{\sqrt{2\pi\sigma(0)^{2}}}\int_{-\infty}^{z}{\rm
d}y\,e^{-\frac{y^{2}}{2\sigma(0)^{2}}}.$ (4.8)

Thus, to start with, as a consistency check we show how to derive (4.8). If we
call, using the notation in (2.19) and setting
$\mu(\psi):=1-\nu(\psi)=1+\rho\partial_{\varphi}f(0,\psi)$,

$W_{0}:=\sum_{i=1}^{\infty}\Biggl{(}\prod_{j=1}^{i-1}\mu{\circ}A_{0}^{-j}\Biggr{)}\beta{\circ}A_{0}^{-i}\,,$

we prove in Subsection 7.3 that $\langle|W-W_{0}|\rangle=O(\rho)$. This
implies that $(W-W_{0})/\sqrt{\rho}$ converges in probability to 0, so that we
just need to prove that $W_{0}/\sqrt{\rho}$, in the limit, has the correct
normal distribution. Consider now the new random variable

$W_{00}(\psi):=\sum_{i=1}^{\infty}\overline{\mu}^{i-1}\beta{\circ}A_{0}^{-i}\,.$

In Appendix E we present a partial extension of Proposition 7.6 to multi-times
correlation functions, which yields that

$\big{\langle}{\left(W_{0}-W_{00}\right)^{2}}\big{\rangle}=O(\rho^{2}).$ (4.9)

On the other hand, the Central Limit Theorem for Anosov system [16] implies
that $W_{00}/\sqrt{\rho}$ tends to the correct normal limit as $\rho\to
0^{+}$.

We expect a similar strategy to work for the derivation of (4.7), by
proceeding along the lines of Subsection 7.8.

###  4.3 Large Deviations

Finally, the discussion in Subsection 4.2 naturally leads us to consider the
validity of a large deviation principle for our model. In particular (4.8)
deals with the fluctuation of order $\sqrt{\rho}$ of $W$ around 0. We can then
ask if we can describe the fluctuations of $W$ of order 1. In its simplest
form we expect that, as $\rho\to 0$,

$-\lim_{\rho\to
0}\rho\log\left(m_{0}\left(\\{\psi\in\mathds{T}^{2}:W(\psi)>x\\}\right)\right)=I(x),$

where $I$, the _large deviation rate_ , should be given by the Legendre
transform of the a suitable limit moment generating function for $W$, that is

$I(x)=\sup_{\tau>0}\left(\tau x-\lambda(\tau)\right),$

with

$\lambda(\tau)=\lim_{\rho\to 0}\rho\log\left(\left\langle e^{\tau
W}\right\rangle\right).$

Given the essentially explicit expression for $W$ contained in Subsection 6.1,
we think it should be possible to show that $I$ exists and to compute it by
using the methods and results of our paper.

It would be more interesting to study the large fluctuation of the full
process $X_{t}$ defined in (2.46). Again, we think that it should be possible
to show that, at fixed $\varphi_{0}$, as $\rho\to 0$,

$-\lim_{\rho\to
0}\rho\log\left(m_{0}\Big{(}\Big{\\{}\psi\in\mathds{T}^{2}:\sup_{t\geq
0}e^{\xi t}|\widetilde{X}_{t}(\psi)-\Phi_{t}|>x\Big{\\}}\Big{)}\right)=J(x),$

where $\widetilde{X}_{t}$ is defined in (2.47) and $\xi\in[0,\gamma)$ is as in
Theorem 8, while $J$ is a large deviation rate to be related to a suitable
moment generating function for the full dynamics.

##  5 Applications to physical problems

###  5.1 Periodic orbits: Krylov-Bogolyubov theory

One of the first problems to be studied, where fast and slow variables are
coupled to each other, were the planetary motions in celestial mechanics. A
well-known example are the effects of the revolution of the Moon around the
Earth (the fast motion) on the revolution of the Earth around the Sun (the
slow motion).

Krylov-Bogolyubov theory provides a useful tool to deal with such a kind of
problem and, more generally, to study the behaviour of oscillating systems
where at least two very different time scales are involved: an averaged
equation for the slow variables is obtained after integrating out the motions
of the fast variables [38, 9, 35]. The theory has been successfully applied to
a wide class of dynamical systems, which range from very simple two-
dimensional systems, such as the Van der Pol equation or the inverted
pendulum, to much more complicated ones, such as the stability of the Solar
system, where, because of the complexity of the equations, numerical analysis
plays a dominant role.

Recently the averaging method has been applied to study the stability of the
$H_{2}^{+}$ ion within the framework of classical mechanics [14]. Integrating
out the electron coordinates, treated as fast variables, leads to an effective
Hamitlonian describing the motion of the two protons. What is found
numerically is that, for certain initial conditions of the electron
coordinates, the protons are captured in an oscillatory state. This can be
seen as a synchronization phenomenon which causes the protons to stabilize on
a suitable periodic orbit. On the other hand, the numerical simulations also
show that for other initial conditions the motion of the electron becomes
chaotic. In this case, apparently no regular pattern emerges for the motion of
the protons. It would be interesting to investigate further the chaotic
regime, in the light of the increasing results in the literature showing that
synchonization may still occur when the dynamics of the fast variable moves
from regular to a chaotic; for instance, a behaviour of this kind is observed
numerically in electromechanical systems with flexible arms (see ref. [39] and
references quoted therein).

###  5.2 Climate models

The problem of climate change has been increasingly investigated recently,
also in the light of its strong relation with society and life on our planet.
Earth’s climate system is undoubtedly one of the most significant examples of
chaotic systems where fast and slow variables interact with each other: while
weather processes, such as the atmospheric and ocean dynamics, can be
considered as fast motions, what one is ultimately interested in is the slow
evolution of Earth’s climate [37]. Moreover, one has to take into account also
intentional and unintentional human-induced perturbations, such as the global
warming due to human activities.

As a consequence of the wide range of processes and external forces involved
in the climate system, the mathematical models which are used to treat the
problem in full generality are inevitably complicated, and the corresponding
differential equations are mainly studied numerically. To attack the problem
analytically, the effects of the small-scale processes are usually taken into
account in the equations governing the dynamics on large scale by introducing
a suitable parameterization, which may be deterministic in some cases but,
more frequently, leads to stochastic differential equations. However, also
analytically more accessible models have been studied, both because there are
problems which admit a simpler description and because obtaining analytical
results allows us to improve our general understanding of the problem. A class
of simple climate models are the _energy balance models_ , where only a few
variables appear. For instance, one can consider a two-dimensional model,
where the evolution of the mean surface temperature and of the mean deep ocean
temperature is governed by a system of two stochastic differential equations:
the climate system response is characterised by two timescales, with the deep
ocean temperature reacting much more slowly [52].

The use of mathematical models in order to deal with the climate change has
intensified in the last few years, thanks to the recent developments in
dynamical systems theory as well as in statistical mechanics and probability;
see for instance refs. [25, 42, 32] for reviews on the topics. The tools we
use in the present paper provide a possible path to follow in order to address
the analysis of climate models. Studying coupled Anosov systems, which in
principle could appear a mathematical abstraction, is justified in
consideration of Gallavotti-Cohen chaotic hypothesis [31]. In this regard, we
stress that the results obtained by relying on Ruelle response theory [43, 32]
exploited the very same assumption.

###  5.3 Several coupled systems: the heat equation

As we already mentioned, a well established line of research aim to a
derivation of the macroscopic law of transport of energy in a crystal, that is
the heat equation, starting from the deterministic microscopic dynamics. In
this spirit one considers a large number $N$ of microscopic systems – which
can be taken equal to each other – organized on a lattice in $\mathds{R}^{3}$.
Without interaction each microscopic system presents a neutral direction that
represents the fact that energy is locally conserved. After a small
interaction that couples the neutral directions is introduced, one expect to
see the heat equation to emerge as an effective macroscopic equation in the
_hydrodynamic limit_ , that is the limit in which both the number $N$ of local
systems and the (discrete) time $k$ go to infinity in such a way that $k\sim
N^{2}$.

An interesting result in this direction is obtained in ref. [12], where the
local systems are assumed to be chaotic maps coupled with a neutral variable
which play the role of a local energy, and a further conservative small
interaction is introduced between the local systems. Then, for initial
conditions with the energies confined in a very small region, a diffusion
equation is proved to be satisfied by the local energies at finite time.

Stated in its full generality, the problem is too hard for our present
knowledge. As discussed in refs. [22, 41], a possible strategy to pursue is to
split the study into two separate steps:

  1. 1.

First one studies a single local system weakly interacting with a neutral
variable in the limit in which the size $\rho$ of the interaction vanishes
(scaling regime). In order to obtain a non-trivial evolution one studies the
behaviour of the system for times $k$ diverging with a law which depends on
$\rho$. In this way one obtains a differential equation (the _mesoscopic
equation_) describing the dynamics of the neutral variable.

  2. 2.

Next, one couples a large number $N$ of such systems and takes the limit in
which both $N$ and the rescaled time go to infinity according to the
hydrodynamic limit. The heat equation should emerge as the partial
differential equation describing the evolution of the local energy
concentration, that is the average of the neutral variable in a small region.

At the moment, we are not aware of substantial progress in respect to the
second step. As far as the first step is concerned, one wishes the dynamics to
be well understood in the absence of interaction. On the other hand,
integrable systems have to be excluded because they are too special and are
expected to display a non-typical behaviour. For these reasons, the local
systems are usually assumed to be chaotic, as in [12] – for instance expanding
maps or, as in our paper, Anosov maps. A further simplifying hypothesis is to
consider an interaction which makes non-conservative the evolution of the
neutral variable, so that local attractors appear: this is required in order
to control the dynamics over long times. In such a situation the scaling limit
requires $k\sim\rho^{-1}$ and the mesoscopic equation is an ordinary
differential equation. In the more difficult conservative case, a different
scaling law $k\sim\rho^{-2}$ is looked for and the mesoscopic equation is
expected to be a stochastic differential equation (see also Subsection 4.2 for
more comments on this point).

The present paper deals with the first step of the strategy outlined above, in
the strictly dissipative case. The next step would be considering a finite
region $\Lambda\subset\mathds{Z}^{d}$ and a set of variable
$(\underline{\varphi},\underline{\psi})\in\mathds{T}^{\Lambda}\times(\mathds{T}^{2})^{\Lambda}$,
such that, for $i\in\Lambda$, the dynamics is given by

${\mathscr{S}}_{i}(\underline{\varphi},\underline{\psi})=(\varphi_{i}+\rho
f_{i}(\underline{\varphi},\underline{\psi}),A_{0}\psi_{i}),$

with $f_{i}(\underline{\varphi},\underline{\psi})$ depending only on the
variables $\varphi_{j}$ and $\psi_{j}$ with $j\in\Lambda$ close to $i$, for
instance the first neighbours. The results of the paper should extend to the
case of a finite number of systems, while extending the analysis to an
arbitrarily large region $\Lambda$ requires substantial additional work in
order to obtain bounds uniform in the size of the region. Of course, because
of the dissipation, one does not expect to obtain the heat equation when the
hydrodynamic limit is taken; nevertheless, studying the limit of infinitely
many systems in a simpler case could shed light on the more realistic models.

As the last comment suggests, another non-trivial extension would be removing
the dissipation hypothesis on the dynamics of the neutral variable. However,
as stressed in ref. [41] this is a much harder problem with respect to the
dissipative case, already in the case of a single system.

##  6 Mapping to a simpler model

In this section we prove Theorems 1 and 2 by explicitly solving (2.21) and
(2.23). The first theorem is obtained by relying on Banach Fixed-Point
Theorem, while the second one exploits the dynamics being uniformly
contracting around the invariant manifold. Finally, the two results together
are showed to yield immediately Theorem 3.

###  6.1 The invariant manifold: proof of Theorem 1

From (2.21) we get

$W(A_{0}\psi)=G(W(\psi),\psi)=W(\psi)+F(W(\psi),\psi)\,.$ (6.1)

To show that a solution of (6.1) exists we define the map

${\mathscr{E}}[W](\psi):=G(W(A_{0}^{-1}\psi),A_{0}^{-1}\psi)$ (6.2)

so that the invariant manifold is the solution of the fixed point equation
${\mathscr{E}}[W]=W$.

If we define

$B:=\\{W\in{\mathfrak{B}}(\mathds{T}^{2},\mathds{R}):S_{m}\leq W\leq
S_{M}\\},$ (6.3)

then ${\mathscr{E}}(B)\subset B$ by Hypotheses 1 and 2. Moreover

$\displaystyle\|{\mathscr{E}}[W_{1}]-{\mathscr{E}}[W_{2}]\|_{\infty}=$
$\displaystyle\sup_{\psi\in\mathds{T}^{2}}\left|G(W_{1}(A_{0}^{-1}\psi),A_{0}^{-1}\psi)-G(W_{2}(A_{0}^{-1}\psi),A_{0}^{-1}\psi)\right|$
$\displaystyle=$
$\displaystyle\sup_{\psi\in\mathds{T}^{2}}\left|G(W_{1}(\psi),\psi)-G(W_{2}(\psi),\psi)\right|\leq(1-\Gamma)\|W_{1}-W_{2}\|_{\infty},$

that is ${\mathscr{E}}$ is a contraction on $B$, and thus, by the Banach
Fixed-Point Theorem, there is a unique $W\in B$ that satisfies (6.1).

To discuss the regularity of $W$ we observe that, since $F$ is
$\alpha_{0}$-Hölder continuous, we get, for $\alpha_{+}\in(0,\alpha_{0}]$,

$\displaystyle|{\mathscr{E}}[W]|_{\alpha_{+}}^{+}$
$\displaystyle=\lambda^{-\alpha_{+}}\sup_{\psi\in\mathds{T}^{2}}\sup_{x\in\mathds{R}}|x|^{-\alpha_{+}}\left|G(W(\psi+xv_{+}),\psi+xv_{+})-G(W(\psi),\psi)\right|$
(6.4)
$\displaystyle\leq\lambda^{-\alpha_{+}}\sup_{\psi\in\mathds{T}^{2}}\sup_{x\in\mathds{R}}|x|^{-\alpha_{+}}\Big{(}|G(W(\psi+xv_{+}),\psi+xv_{+})-G(W(\psi),\psi+xv_{+})|$
$\displaystyle\qquad\qquad\qquad+|G(W(\psi),\psi+xv_{+})-G(W(\psi),\psi)|\Bigr{)}$
$\displaystyle\leq\lambda^{-\alpha_{+}}\left((1-\Gamma)|W|_{\alpha_{+}}^{+}+|F|_{\alpha_{+}}\right),$

and similarly, for $\alpha_{-}\in(0,\alpha_{0}]$,

$|{\mathscr{E}}[W]|_{\alpha_{-}}^{-}\leq\lambda^{\alpha_{-}}\left((1-\Gamma)|W|_{\alpha_{-}}^{-}+|F|_{\alpha_{-}}\right)\,.$
(6.5)

This implies that the set

$B_{\alpha_{-},\alpha_{+}}:=\left\\{W\in{\mathfrak{B}}_{\alpha_{-},\alpha_{+}}(\mathds{T}^{2},\mathds{R}):|W|_{\alpha_{+}}^{+}\leq\frac{|F|_{\alpha_{+}}}{\lambda^{\alpha_{+}}-(1-\Gamma)}\;,\quad|W|_{\alpha_{-}}^{-}\leq\frac{|F|_{\alpha_{-}}}{\lambda^{-\alpha_{-}}-(1-\Gamma)}\right\\}$
(6.6)

is invariant under ${\mathscr{E}}$. Thus, we need to show that ${\mathscr{E}}$
is a contraction on $B_{\alpha_{-},\alpha_{+}}$ for suitable $\alpha_{-}$ and
$\alpha_{+}$, in order to apply once more the Banach Fixed-Point Theorem. To
this end, observe that

$\displaystyle|{\mathscr{E}}[W_{1}]-{\mathscr{E}}[W_{2}]|_{\alpha_{+}}^{+}$
$\displaystyle=\lambda^{-\alpha_{+}}\sup_{\psi\in\mathds{T}^{2}}\sup_{x\in\mathds{R}}|x|^{-\alpha_{+}}\Bigl{|}G(W_{1}(\psi),\psi)-G(W_{2}(\psi),\psi)$
(6.7)
$\displaystyle\qquad\qquad-G(W_{1}(\psi+xv_{+}),\psi+xv_{+})+G(W_{2}(\psi+xv_{+}),\psi+xv_{+})\Bigr{|}$
$\displaystyle\leq\lambda^{-\alpha_{+}}\sup_{\psi\in\mathds{T}^{2}}\sup_{x\in\mathds{R}}|x|^{-\alpha_{+}}\Bigl{|}G(W_{1}(\psi),\psi)-G(W_{2}(\psi),\psi)$
$\displaystyle\qquad\qquad-G(W_{1}(\psi+xv_{+}),\psi)+G(W_{2}(\psi+xv_{+}),\psi)\Bigr{|}\phantom{\sup_{x}}$
$\displaystyle+\lambda^{-\alpha_{+}}\sup_{\psi\in\mathds{T}^{2}}\sup_{x\in\mathds{R}}|x|^{-\alpha_{+}}\Bigl{|}G(W_{1}(\psi+xv_{+}),\psi)-G(W_{2}(\psi+xv_{+}),\psi)$
$\displaystyle\qquad\qquad-G(W_{1}(\psi+xv_{+}),\psi+xv_{+})+G(W_{2}(\psi+xv_{+}),\psi+xv_{+})\Bigr{|}\,.$

Writing

$G(W_{1}(\psi_{1}),\psi_{2})-G(W_{2}(\psi_{1}),\psi_{2})\\!=\\!\int_{0}^{1}\\!\\!{\rm
d}t\,\partial_{\varphi}G(tW_{1}(\psi_{1})+(1-t)W_{2}(\psi_{1}),\psi_{2})(W_{1}(\psi_{1})-W_{2}(\psi_{1})),$

we bound, in the contribution from the fifth and sixth lines in (6.7),

$\displaystyle\sup_{\psi\in\mathds{T}^{2}}\sup_{x\in\mathds{R}}|x|^{-\alpha_{+}}\Bigl{|}G(W_{1}(\psi+xv_{+}),\psi)-G(W_{2}(\psi+xv_{+}),\psi)$
(6.8)
$\displaystyle\quad-G(W_{1}(\psi+xv_{+}),\psi+xv_{+})+G(W_{2}(\psi+xv_{+}),\psi+xv_{+})\Bigr{|}\leq|\partial_{\varphi}G|_{\alpha_{+}}\|W_{1}-W_{2}\|_{\infty}.$

For any $C^{2}$ function $h$ we have

$\displaystyle h(x_{1})-h(x_{2})-h(x_{3})+h(x_{4})$
$\displaystyle\qquad=\frac{1}{2}\int_{0}^{1}{\rm
d}t\,\bigl{(}h^{\prime}(tx_{1}+(1-t)x_{2})+h^{\prime}(tx_{3}+(1-t)x_{4})\bigr{)}(x_{1}-x_{2}-x_{3}+x_{4})$
$\displaystyle\qquad+\frac{1}{2}(x_{1}-x_{2}+x_{3}-x_{4})\int_{0}^{1}{\rm
d}t\,(tx_{1}+(1-t)x_{2}-tx_{3}-(1-t)x_{4})$
$\displaystyle\phantom{\int_{0}^{1}\int_{0}^{1}dtds}\times\int_{0}^{1}{\rm
d}s\,h^{\prime\prime}\bigl{(}s(tx_{1}+(1-t)x_{2})+(1-s)(tx_{3}+(1-t)x_{4})\bigl{)},$

so that we can bound, in the contribution from the third and fourth lines of
(6.7),

$\displaystyle\sup_{\psi\in\mathds{T}^{2}}\sup_{x\in\mathds{R}}|x|^{-\alpha_{+}}\Bigl{|}G(W_{1}(\psi),\psi)-G(W_{2}(\psi),\psi)-G(W_{1}(\psi+xv_{+}),\psi)$
(6.9)
$\displaystyle\qquad+G(W_{2}(\psi+xv_{+}),\psi)\Bigr{|}\leq(1-\Gamma)|W_{1}-W_{2}|_{\alpha_{+}}^{+}+\|\partial_{\varphi}^{2}G\|_{\infty}\max_{i=1,2}|W_{i}|_{\alpha_{+}}^{+}\,\|W_{1}-W_{2}\|_{\infty}.$

Collecting the bounds (6.8) and (6.9), we obtain

$\displaystyle|{\mathscr{E}}[W_{1}]-{\mathscr{E}}[W_{2}]|_{\alpha_{+}}^{+}$
(6.10)
$\displaystyle\qquad\leq\lambda^{-\alpha_{+}}\Bigl{(}(1-\Gamma)|W_{1}-W_{2}|_{\alpha_{+}}^{+}+\bigl{(}\|\partial_{\varphi}^{2}F\|_{\infty}\max_{i=1,2}|W_{i}|_{\alpha_{+}}^{+}+|\partial_{\varphi}F|_{\alpha_{+}}^{+}\bigr{)}\|W_{1}-W_{2}\|_{\infty}\Bigr{)}$

and, analogously,

$\displaystyle|{\mathscr{E}}[W_{1}]-{\mathscr{E}}[W_{2}]|_{\alpha_{-}}^{-}$
(6.11)
$\displaystyle\qquad\leq\lambda^{\alpha_{-}}\Bigl{(}(1-\Gamma)|W_{1}-W_{2}|_{\alpha_{-}}^{-}+\bigl{(}\|\partial_{\varphi}^{2}F\|_{\infty}\max_{i}|W_{i}|_{\alpha_{-}}^{-}+|\partial_{\varphi}F|_{\alpha_{-}}^{-}\bigr{)}\|W_{1}-W_{2}\|_{\infty}\Bigr{)}.$

Let now $\alpha_{-},\alpha_{+}\in(0,\alpha_{0}]$ be such that

$\displaystyle\lambda^{\alpha_{-}}(1-\Gamma)<1\,,\phantom{\sum_{\sum}}$
(6.12a)
$\displaystyle\alpha_{+}\lambda^{-\alpha_{+}}\\!\Biggl{(}\frac{\|\partial_{\varphi}^{2}F\|_{\infty}|F|_{\alpha_{+}}}{\lambda^{\alpha_{+}}-(1-\Gamma)}+|\partial_{\varphi}F|_{\alpha_{+}}^{+}\Biggr{)}+\alpha_{-}\lambda^{\alpha_{-}}\\!\Biggl{(}\frac{\lambda^{\alpha_{-}}\|\partial_{\varphi}^{2}F\|_{\infty}|F|_{\alpha_{-}}}{1-\lambda^{\alpha_{-}}(1-\Gamma)}+|\partial_{\varphi}F|_{\alpha_{-}}^{-}\Biggr{)}\\!<\Gamma\,.$
(6.12b)

For such $\alpha_{-}$ and $\alpha_{+}$, combining the bounds (6.4), (6.5),
(6.10) and (6.11), the map ${\mathscr{E}}$ turns out to be a contraction on
$B_{\alpha_{-},\alpha_{+}}$. This concludes the proof of Theorem 1.

######  Remark 6.1.

_Recalling that $S_{m}<0<S_{M}$, by Remark 2.10, the discussion above implies
that the sequence ${\mathscr{E}}^{n}[0]$ converges to $W$ in
${\mathfrak{B}}(\mathds{T}^{2},\mathds{R})$. _

######  Remark 6.2.

_The inequalities ( 6.12) are satisfied for $\alpha_{-},\alpha_{+}$ small
enough. If $\rho$ is small and
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|F|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{0},2}=O(\rho)$,
then (6.12a) requires $\alpha_{-}=O(\rho)$, which inserted into (6.12b) shows
that $\alpha_{+}$ is allowed to be $O(1)$ in $\rho$. _

######  Remark 6.3.

_Assuming the map ${\mathscr{S}}$ to satisfy much stronger regularity
properties in the fast variable (such as smoothness or even analyticity) does
not really improve the regularity of the manifold. At best, we might obtain a
regularity somewhat stronger than the Hölder continuity, such as the strong
Hölder condition considered in ref. [6], but only at the price of assuming
stronger conditions on the map, in particular on the variation of the function
$\mu$ – see also Remark 2.19. _

###  6.2 The invariant manifold in the scaling regime: proof of Lemma 2.21

In the scaling regime, where $F$, together with all its derivatives, is
proportional to $\rho$, the best we can say about the invariant manifold is
that $W\in B$, with $B$ as in (6.3), and hence
$|W|\leq\max\\{|S_{m}|,S_{M}\\}$, so that we still have $\|W\|_{\infty}=O(1)$.
Furthermore, since $W\in B_{\alpha_{-},\alpha_{+}}$, with
$B_{\alpha_{-},\alpha_{+}}$ as in (6.6), so that $\alpha_{+}=O(1)$ and
$\alpha_{-}=O(\rho)$ by Remark 6.2, we obtain $|W|_{\alpha_{-}}^{-}=O(1)$,
while $|W|_{\alpha_{+}}^{+}=O(\rho)$.

######  Remark 6.4.

_If $\psi_{0}$ is a fixed point of $A_{0}$, that is $A_{0}\psi_{0}=\psi_{0}$,
we get $W(\psi_{0})=S(\psi_{0})$, so that, in the scaling regime,
$W(\psi_{0})$ does not depend on $\rho$. In a similar way, if $\psi_{k}$ is a
periodic point of period $k$, that is $A_{0}^{k}\psi_{k}=\psi_{k}$, then
$W(\psi_{k})$ is found between $\min_{i}S(A_{0}^{i}\psi_{k})$ and
$\max_{i}S(A_{0}^{i}\psi_{k})$. This shows that, in the scaling regime, in
general $\lim_{\rho\to 0^{+}}W(\psi)\not=0$. _

###  6.3 The conjugation: proof of Theorem 2

####  6.3.1 The translated map

We will construct the conjugation ${\mathscr{H}}$ by first subtracting the
steady state and then linearizing the resulting dynamics around 0. More
precisely, we write $\varphi=\theta+W(\psi)$ so that, in terms of the
variables $(\theta,\psi)$, the dynamics is described by the map

${\mathscr{S}}_{1}(\theta,\psi)=(({\mathscr{S}}_{1})_{\theta}(\theta,\psi),({\mathscr{S}}_{1})_{\psi}(\theta,\psi)):=(G_{1}(\theta,\psi),A_{0}\psi),\qquad
G_{1}(\theta,\psi):=\theta+F_{1}(\theta,\psi),$ (6.13)

with

$F_{1}(\theta,\psi):=F(\theta+W(\psi),\psi)-F(W(\psi),\psi)=F(\theta+W(\psi),\psi)+W(\psi)-W(A_{0}\psi).$
(6.14)

We call ${\mathscr{S}}_{1}$ the _translated map_. Defining

$\Omega_{1}:=\\{(\theta,\psi)\in\mathds{T}\times\mathds{T}^{2}\,:\,\phi_{m}-W(\psi)\leq\theta\leq\phi_{M}-W(\psi)\\},$
(6.15)

we have that ${\mathscr{S}}_{1}$ is injective from $\Omega_{1}$ into itself.

######  Remark 6.5.

_One easily checks that ${\mathscr{S}}_{1}(0,\psi)=(0,A_{0}\psi)$, so that,
expressed in terms of $\theta$, the invariant manifold reduces to
$\overline{\mathcal{W}}=\\{(0,\psi):\psi\in\mathds{T}^{2}\\}$ (see Remark
2.26). _

######  Remark 6.6.

_The iterations of the maps ${\mathscr{S}}$ and ${\mathscr{S}}_{1}$ are such
that_

$({\mathscr{S}}^{n})_{\varphi}(\varphi,\psi)=({\mathscr{S}}_{1}^{n})_{\theta}(\varphi-W(\psi),\psi)+W(A_{0}^{n}\psi),$
(6.16)

_while
$({\mathscr{S}}^{n})_{\psi}(\varphi,\psi)=({\mathscr{S}}_{1}^{n})_{\psi}(\varphi-W(\psi),\psi)=A_{0}^{n}\psi$.
Conversely we have_

$({\mathscr{S}}_{1}^{n})_{\theta}(\theta,\psi)=({\mathscr{S}}^{n})_{\varphi}(\theta+W(\psi),\psi)-W(A_{0}^{n}\psi).$

In Subsections 6.3.2 and 6.3.3 we will study the conjugation relation

${\mathscr{H}}_{1}{\circ}{\mathscr{S}}_{1}={\mathscr{S}}_{0}{\circ}{\mathscr{H}}_{1}.$
(6.17)

where ${\mathscr{H}}_{1}:\Omega_{1}\to\Omega_{0}$ is of the form
${\mathscr{H}}_{1}(\theta,\psi)=({\mathcal{H}}_{1}(\theta,\psi),\psi)$. We can
then write

${\mathscr{H}}(\varphi,\psi)=({\mathcal{H}}_{1}(\varphi-W(\psi),\psi),\psi),$
(6.18)

with ${\mathscr{H}}$ satisfying the conjugation relation (2.23). Thus, if the
conjugation ${\mathscr{H}}_{1}$ exists and is invertible, the conjugation
${\mathscr{H}}$ exists and is invertible as well, and vice versa. Moreover,
${\mathscr{H}}$ and ${\mathscr{H}}_{1}$ have the same image $\Omega_{0}$
defined in Theorem 2. In analogy with (6.18), we write
${\mathscr{H}}_{1}^{-1}(\eta,\psi)=({\mathcal{L}}_{1}(\eta,\psi),\psi)$.

Considering also Remark 2.14, we look for functions
${\mathcal{H}}_{1}\\!:\Omega_{1}\to\mathds{R}$ and
${\mathcal{L}}_{1}:\Omega_{0}\to\mathds{R}$ of the form

${\mathcal{H}}_{1}(\theta,\psi)=\theta+\theta^{2}h_{1}(\theta,\psi),\qquad{\mathcal{L}}_{1}(\eta,\psi)=\eta+\eta^{2}l_{1}(\eta,\psi).$
(6.19)

If we show that functions ${\mathcal{H}}_{1}$ and ${\mathcal{L}}_{1}$ of the
form (6.19) exist, with
$h_{1}\in{\mathscr{B}}_{\alpha_{*},1}(\Omega_{1},\mathds{R})$ and
$l_{1}\in{\mathscr{B}}_{\alpha_{*},1}(\Omega_{0},\mathds{R})$, for a suitable
$\alpha_{*}\in(0,\alpha_{0})$, then we can write the conjugation
${\mathscr{H}}$ and its inverse as in (2.25), with

$h(\varphi,\psi)=h_{1}(\varphi-W(\psi),\psi),\qquad
l(\eta,\psi)=l_{1}(\eta,\psi).$ (6.20)

######  Remark 6.7.

_As a consequence of Theorem 1, there exists a closed interval
$\Theta:=[\theta_{-},\theta_{+}]$, such that
$\Theta\times\mathds{T}^{2}\subset\Omega_{1}$ and hence
$\\{(\varphi,\psi)\in\mathds{T}\times\mathds{T}^{2}\,:\,\varphi-W(\psi)\in\Theta\\}\subset\Omega$._

####  6.3.2 Dynamics toward the steady state

From (6.17) we see that ${\mathcal{H}}_{1}$ satisfies the equation

$\kappa(\psi){\mathcal{H}}_{1}(\theta,\psi)={\mathcal{H}}_{1}({\mathscr{S}}_{1}(\theta,\psi))\,.$
(6.21)

According to (6.19), we get

$h_{1}(\theta,\psi)=\frac{1}{\kappa(\psi)\theta^{2}}\Bigl{(}G_{1}(\theta,\psi)+G_{1}(\theta,\psi)^{2}h_{1}({\mathscr{S}}_{1}(\theta,\psi))-\kappa(\psi)\,\theta\Bigr{)},$

with $G_{1}(\theta,\psi)$ as in (6.13), so that, setting

$\displaystyle p_{1}(\theta,\psi):=$
$\displaystyle\frac{(G_{1}(\theta,\psi))^{2}}{\theta^{2}\kappa(\psi)}=\frac{1}{\kappa(\psi)}\left(1+\int_{0}^{1}{\rm
d}t\,\partial_{\varphi}F(t\theta+W(\psi),\psi)\right)^{2},$ (6.22a)
$\displaystyle q_{1}(\theta,\psi):=$
$\displaystyle\frac{G_{1}(\theta,\psi)-\theta\kappa(\psi)}{\theta^{2}\kappa(\psi)}=\frac{1}{\kappa(\psi)}\int_{0}^{1}{\rm
d}t\,(1-t)\,\partial_{\varphi}^{2}F(t\theta+W(\psi),\psi)\,,$ (6.22b)

we obtain

$h_{1}(\theta,\psi)=q_{1}(\theta,\psi)+p_{1}(\theta,\psi)\,h_{1}({\mathscr{S}}_{1}(\theta,\psi))\,,$
(6.23)

whose solution can be formally written as

$h_{1}(\theta,\psi)=\sum_{n=1}^{\infty}p_{1}^{(n)}(\theta,\psi)\,q_{1}({\mathscr{S}}_{1}^{n}(\theta,\psi)),$
(6.24)

with

$p^{(n)}_{1}(\theta,\psi):=\prod_{i=0}^{n-1}p_{1}({\mathscr{S}}_{1}^{i}(\theta,\psi))$
(6.25)

where, according to our conventions (see Remark 2.32),
$p^{(0)}_{1}(\theta,\psi)=1$.

######  Remark 6.8.

_Both $p_{1}(\theta,\psi)$ and $q_{1}(\theta,\psi)$ are well defined at
$\theta=0$: using (6.1) and the definition of $\kappa(\psi)$ after (2.22), we
find $\kappa(\psi)\,q_{1}(0,\psi)=\partial_{\varphi}^{2}F(0,\psi)$ and
$p_{1}(0,\psi)=\kappa(\psi)$. _

We start with the regularity properties of the functions $p_{1}$ and $q_{1}$.
The following result is an easy consequence of the representation in (6.22)
together with Lemma 2.9.

######  Lemma 6.9.

Assume the map ${\mathscr{S}}$ in (2.15) to satisfy Hypotheses 1–3. For any
$\Gamma^{\prime}\in(0,\Gamma)$ and any
$\alpha\in(0,\min\\{\alpha_{-},\alpha_{+}\\})$, with $\alpha_{-}$ and
$\alpha_{+}$ as in Theorem 1, one has
$p_{1},q_{1}\in{\mathscr{B}}_{\alpha,5}(\Omega_{1},\mathds{R})$ and
$p_{1}(\theta,\psi)\leq 1-\Gamma^{\prime}$ as long as
$|\theta|\leq\theta_{1}$, with

$\theta_{1}:=\frac{\Gamma-\Gamma^{\prime}}{2\|\kappa^{-1}\|_{\infty}(1+\|\partial_{\varphi}F\|_{\infty})\|\partial_{\varphi}^{2}F\|_{\infty}}.$
(6.26)

_Proof._ Observe that $p_{1}(0,\psi)=1+\partial_{\varphi}F(W(\psi),\psi)\leq
1-\Gamma$. Thus, using that, for any $\theta_{1}>0$ and all
$\theta\in[-\theta_{1},\theta_{1}]$, we have
$|p_{1}(\theta,\psi)-p_{1}(0,\psi)|\leq\|\partial_{\theta}p_{1}\|_{\infty}\theta_{1}$,
if we bound $\partial_{\theta}p_{1}$ using (6.22a) and fix $\theta_{1}$ as in
(6.26), the bound on $p_{1}(\theta,\psi)$ for $|\theta|\leq\theta_{1}$ follows
immediately. The other bounds too are easily obtained by estimating the
derivatives of $p_{1}$ and $q_{1}$. ∎

######  Remark 6.10.

_In the scaling regime, where $\Gamma=\rho\,\gamma$, one has
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|p_{1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,5}=1+O(\rho)$,
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|q_{1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha,5}=O(\rho)$,
while
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}p_{1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,4}=O(\rho)$
and $|p_{1}|_{\alpha}=O(\rho)$. Moreover, if one takes
$\Gamma^{\prime}=\rho\,\gamma^{\prime}$, with $\gamma^{\prime}\in(0,\gamma)$,
then (6.26) in Lemma 6.9 implies that $\theta_{1}=O(1)$. _

Next we study the regularity of the iterates of ${\mathscr{S}}_{1}$.

######  Lemma 6.11.

Assume the map ${\mathscr{S}}$ in (2.15) to satisfy Hypotheses 1–3. For any
$\Gamma^{\prime}\in(0,\Gamma)$ and
$\alpha\in(0,\min\\{\alpha_{-},\alpha_{+}\\})$, with $\alpha_{-}$ and
$\alpha_{+}$ as in Theorem 1, and for all $n\geq 0$, one has
$({\mathscr{S}}^{n}_{1})_{\theta}\in{\mathscr{B}}_{\alpha,2}(\Omega_{1},\mathds{R})$
and

$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|\partial_{\theta}({\mathscr{S}}^{n}_{1})_{\theta}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,1}\leq
C_{0}\left(1-\Gamma^{\prime}\right)^{n},\qquad|({\mathscr{S}}^{n}_{1})_{\theta}|_{\alpha}\leq
C_{0}\lambda^{\alpha
n},\qquad|\partial_{\theta}({\mathscr{S}}^{n}_{1})_{\theta}|_{\alpha}\leq
C_{0}\lambda^{\alpha n},$

with the constant $C_{0}$ depending on $F$ and $\Gamma^{\prime}$ but not on
$n$.

###### Proof.

For a fixed $\Gamma^{\prime}\in(0,\Gamma)$, let $r=r(\Gamma^{\prime})$ and
$N_{r}$ be defined as in Lemma 2.9, and observe that
$N_{r}=O((\Gamma^{\prime})^{-1})$ in such a case. We have
$\|\partial_{\theta}G_{1}\|_{\infty}\leq 1+\|\partial_{\varphi}F\|_{\infty}$,
while for $n\geq N_{r}$ we may bound

$\|(\partial_{\theta}G_{1}){\circ}{\mathscr{S}}^{n}_{1}\|_{\infty}\leq\|(1+\partial_{\varphi}F){\circ}{\mathscr{S}}^{n}\|_{\infty}\leq(1-\Gamma^{\prime}).$

Thus, noting that
$({\mathscr{S}}^{n}_{1})_{\theta}=G_{1}{\circ}{\mathscr{S}}^{n-1}_{1}$, we get

$\|\partial_{\theta}({\mathscr{S}}^{n}_{1})_{\theta}\|_{\infty}=\|\partial_{\theta}(G_{1}{\circ}{\mathscr{S}}^{n-1}_{1})\|_{\infty}\leq\prod_{i=0}^{n-1}\|(\partial_{\theta}G_{1}){\circ}{\mathscr{S}}^{i}_{1}\|_{\infty}\leq
C_{1}\left(1-\Gamma^{\prime}\right)^{n},$ (6.27)

with

$C_{1}:=\left(\frac{1+\|\partial_{\varphi}F\|_{\infty}}{1-\Gamma^{\prime}}\right)^{N_{r}}.$
(6.28)

Moreover, for $n\geq 2$, by (2.7) and (2.12), we have

$|G_{1}{\circ}{\mathscr{S}}^{n-1}_{1}|_{\alpha}\leq\|(\partial_{\theta}G_{1}){\circ}{\mathscr{S}}^{n-1}_{1}\|_{\infty}|G_{1}{\circ}{\mathscr{S}}^{n-2}_{1}|_{\alpha}+\lambda^{\alpha(n-1)}|G_{1}|_{\alpha},$
(6.29)

where
$|G_{1}|_{\alpha}\leq\|\partial_{\varphi}F\|_{\infty}|W|_{\alpha}+|F|_{\alpha}$,
and hence, iterating, we get

$\displaystyle|({\mathscr{S}}^{n}_{1})_{\theta}|_{\alpha}$
$\displaystyle=|G_{1}{\circ}{\mathscr{S}}^{n-1}_{1}|_{\alpha}\leq|G_{1}|_{\alpha}\sum_{i=1}^{n}\lambda^{\alpha(n-i)}\prod_{j=1}^{i-1}\|(\partial_{\theta}G_{1}){\circ}{\mathscr{S}}^{n-j}_{1}\|_{\infty}$
(6.30) $\displaystyle\leq C_{1}|G_{1}|_{\alpha}\,\lambda^{\alpha
n}\sum_{i=1}^{n}\left(1-\Gamma^{\prime}\right)^{i-1}\lambda^{-\alpha i}\leq
C_{2}\lambda^{\alpha n},$

with $C_{1}$ as in (6.28) and

$C_{2}:=C_{1}|G_{1}|_{\alpha}\,(\Gamma^{\prime})^{-1}.$ (6.31)

We have also

$\|\partial_{\theta}^{2}({\mathscr{S}}^{n}_{1})_{\theta}\|_{\infty}\leq\sum_{i=0}^{n-1}\left\|(\partial_{\theta}^{2}G_{1}){\circ}{\mathscr{S}}_{1}^{i}\right\|_{\infty}\left\|\partial_{\theta}({\mathscr{S}}^{i}_{1})_{\theta}\right\|_{\infty}\prod_{\begin{subarray}{c}j=0\\\
j\neq
i\end{subarray}}^{n-1}\|(\partial_{\theta}G_{1}){\circ}{\mathscr{S}}^{j}_{1}\|_{\infty},$

so that, by exploiting (6.27), we find that

$\|\partial_{\theta}^{2}({\mathscr{S}}^{n}_{1})_{\theta}\|_{\infty}\leq
C_{3}\left(1-\Gamma^{\prime}\right)^{n},$

with

$C_{3}:=C_{1}\|\partial_{\varphi}^{2}F\|_{\infty}\,(\Gamma^{\prime})^{-1}.$
(6.32)

Finally, noting that

$\left|\partial_{\theta}({\mathscr{S}}^{n}_{1})_{\theta}\right|_{\alpha}\leq\sum_{i=0}^{n-1}\Bigl{(}\left\|\partial_{\theta}^{2}G_{1}\right\|_{\infty}\left|({\mathscr{S}}^{i}_{1})_{\theta}\right|_{\alpha}+\lambda^{\alpha
i}|\partial_{\theta}G_{1}|_{\alpha}\Bigr{)}\prod_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{n-1}\|(\partial_{\theta}G_{1}){\circ}{\mathscr{S}}^{j}_{1}\|_{\infty},$

and proceeding as done to get the bound (6.30), we obtain

$|\partial_{\theta}({\mathscr{S}}^{n}_{1})_{\theta}|_{\alpha}\leq
C_{4}\lambda^{\alpha n},$

with

$C_{4}:=\bigl{(}C_{2}\|\partial_{\varphi}^{2}F\|_{\infty}+|\partial_{\theta}G_{1}|_{\alpha}\bigr{)}\,(\Gamma^{\prime})^{-1}.$
(6.33)

Then the bounds follow with $C_{0}=\max\\{C_{1},C_{2},C_{3},C_{4}\\}$. ∎

######  Remark 6.12.

_In the scaling regime, where $\Gamma=\rho\,\gamma$, if we choose
$\Gamma^{\prime}=\rho\gamma^{\prime}$, with $\gamma^{\prime}\in(0,\gamma)$, we
find that the constant $C_{0}$ is $O(1)$ in $\rho$. Indeed both
$|{\hskip-0.85355pt}|{\hskip-0.85355pt}|F|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}$
and $|F|_{\alpha}$, and hence $|G_{1}|_{\alpha}$ as well, are $O(\rho)$,
while, in (6.28), one has $N_{r}=O(\rho^{-1})$ by (2.18). _

######  Remark 6.13.

_Using $({\mathscr{S}}^{n}_{1})_{\theta}(0,\psi)=0$ allows us to bound also
$|({\mathscr{S}}^{n}_{1})_{\theta}(\theta,\psi)|\leq\|\partial_{\theta}({\mathscr{S}}^{n}_{1})_{\theta}\|_{\infty}|\theta|$
and hence_

$\|({\mathscr{S}}^{n}_{1})_{\theta}\|_{\infty}\leq
C_{1}\Bigl{(}\max\\{|\phi_{m}|,\phi_{M}\\}+\max\\{|S_{m}|,S_{M}\\}\Bigr{)}(1-\Gamma^{\prime})^{n}.$

####  6.3.3 Existence and regularity of the conjugation

We now have all the ingredients to complete the proof of Theorem 2. Fix
$\Gamma^{\prime}\in(0,\Gamma)$ and set $r=r(\Gamma^{\prime})$, with the
notation of Lemma 2.9. Fix $\theta_{1}$ be as in Lemma 6.9, and set

$S_{r}:=\max\\{S_{M},|S_{m}|\\}+r,\qquad
M_{r}:=\max\\{2\,(\Gamma^{\prime})^{-1}\log(S_{r}/\theta_{1}),0\\}.$

If $(\theta+W(\psi),\psi)\in\Lambda_{r}$ and $n>M_{r}$, then
$|({\mathscr{S}}_{1}^{n}(\theta,\psi))_{\theta}|\leq\theta_{1}$. Combining the
latter with property 3 in Lemma 2.9, for every $(\theta,\psi)\in\Omega_{1}$ we
find that $|({\mathscr{S}}_{1}^{n}(\theta,\psi))_{\theta}|\leq\theta_{1}$ for
$n\geq M_{r}^{\prime}:=M_{r}+N_{r}$. This implies that, for $n\geq
M_{r}^{\prime}$,

$\bigl{\|}p_{1}^{(n)}\bigr{\|}_{\infty}\leq\|p_{1}\|_{\infty}^{M_{r}^{\prime}}\left(1-\Gamma^{\prime}\right)^{n-M_{r}^{\prime}},$
(6.34)

with $p_{1}^{(n)}$ as in (6.25), and hence, using the bound (6.34) in (6.24),
we obtain

$\|h_{1}\|_{\infty}\leq
D_{1}\sum_{n=0}^{\infty}\left(1-\Gamma^{\prime}\right)^{n}\|q_{1}\|_{\infty}\leq
D_{1}(\Gamma^{\prime})^{-1}\|q_{1}\|_{\infty},\qquad\qquad
D_{1}:=\left(\frac{\|p_{1}\|_{\infty}}{1-\Gamma^{\prime}}\right)^{M_{r}^{\prime}}.$
(6.35)

Furthermore, using Lemma 6.11, we see that

$\displaystyle\bigl{\|}\partial_{\theta}p_{1}^{(n)}\bigr{\|}_{\infty}$
$\displaystyle\leq\sum_{i=0}^{n-1}\bigl{\|}\partial_{\theta}\bigl{(}p_{1}{\circ}{\mathscr{S}}_{1}^{i}\bigr{)}\bigr{\|}_{\infty}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\bigl{\|}p_{1}{\circ}{\mathscr{S}}_{1}^{j}\bigr{\|}_{\infty}$
(6.36) $\displaystyle\leq
C_{0}D_{1}\|\partial_{\theta}p_{1}\|_{\infty}\left(1-\Gamma^{\prime}\right)^{n-1}\sum_{i=0}^{n-1}\left(1-\Gamma^{\prime}\right)^{i}\leq
C_{0}D_{1}(\Gamma^{\prime})^{-1}\|\partial_{\theta}p_{1}\|_{\infty}\left(1-\Gamma^{\prime}\right)^{n}\,,$

with $D_{1}$ as in (6.35), so that, summing over $n$, we get

$\|\partial_{\theta}h_{1}\|_{\infty}\leq\sum_{n=1}^{\infty}\left(\bigl{\|}\partial_{\theta}p_{1}^{(n)}\bigr{\|}_{\infty}\|q_{1}\|_{\infty}+\bigl{\|}p_{1}^{(n)}\bigr{\|}_{\infty}\|\partial_{\theta}(q_{1}{\circ}{\mathscr{S}}_{1}^{n})\|_{\infty}\right)\leq
D_{2}\,(\Gamma^{\prime})^{-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|q|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,1}\,,$
(6.37)

for a suitable constant $D_{2}$. By studying along the same lines the second
derivative $\partial_{\theta}^{2}h_{1}$, we obtain (see Appendix B.1 for
details)

$\|\partial_{\theta}^{2}h_{1}\|_{\infty}\leq
D_{3}\,(\Gamma^{\prime})^{-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|q|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{0,2}\,,$
(6.38)

for some other constant $D_{3}$.

Finally, for any $\alpha\in(0,\min\\{\alpha_{-},\alpha_{+}\\}]$, we get, again
relying on Lemma 6.11,

$\displaystyle\bigl{|}p^{(n)}_{1}\bigr{|}_{\alpha}$
$\displaystyle\leq\sum_{i=0}^{n-1}\bigl{|}p_{1}{\circ}{\mathscr{S}}_{1}^{i}\bigr{|}_{\alpha}\prod_{\begin{subarray}{c}j=0\\\
j\not=i\end{subarray}}^{n-1}\bigl{\|}p_{1}{\circ}{\mathscr{S}}_{1}^{j}\bigr{\|}_{\infty}$
(6.39) $\displaystyle\leq
C_{0}D_{1}(|p_{1}|_{\alpha}+\|\partial_{\theta}p_{1}\|_{\infty})\left(1-\Gamma^{\prime}\right)^{n-1}\frac{\lambda^{\alpha
n}-1}{\lambda^{\alpha}-1}\leq D_{4}\,\lambda^{\alpha
n}\left(1-\Gamma^{\prime}\right)^{n}\,,$

for a suitable constant $D_{4}$ proportional to $D_{1}$. To sum over $n$ we
take $\alpha=\alpha_{*}$ in Lemma 6.11, with $\alpha_{*}$ such that
$\lambda^{\alpha_{*}}(1-\Gamma^{\prime})<1-\Gamma^{\prime\prime}$, with
$\Gamma^{\prime\prime}<\Gamma^{\prime}$, so as to obtain

$|h_{1}|_{\alpha_{*}}\leq\sum_{n=1}^{\infty}\left(\bigl{|}p_{1}^{(n)}\bigr{|}_{\alpha_{*}}\|q\|_{\infty}+\bigl{\|}p_{1}^{(n)}\bigr{\|}_{\infty}|q_{1}{\circ}{\mathscr{S}}_{1}^{n}|_{\alpha_{*}}\right)\leq
D_{4}\,(\Gamma^{\prime\prime})^{-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|q_{1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{*},1},$
(6.40)

for a suitable constant $D_{4}$ proportional to $D_{1}$. Once more we bound
$|\partial_{\theta}h_{1}|_{\alpha_{*}}$ by reasoning in a similar way (again
we refer to Appendix B.1 for details) and find

$|\partial_{\theta}h_{1}|_{\alpha_{*}}\leq
D_{5}\,(\Gamma^{\prime\prime})^{-1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|q_{1}|{\hskip-0.85355pt}|{\hskip-0.85355pt}|_{\alpha_{*},1},$
(6.41)

with $D_{5}$ proportional to $D_{2}$.

By collecting together the bounds (6.35), (6.37), (6.38), (6.40) and (6.41),
we find that $\|h_{1}\|_{\alpha_{*},2}$ is bounded. Therefore, by recalling
the first relation in (6.20), Theorem 2 is proved, with
$\Omega_{0}={\mathscr{H}}_{1}(\Omega_{1})={\mathscr{H}}(\Omega)$.

######  Remark 6.14.

_The argument above show that, essentially, it is enough to prove the
existence of the conjugation inside $\Lambda$. Indeed, once the conjugation
has been defined in $\Lambda$, it can be easily extended to the whole $\Omega$
by using the fact that all trajectories fall inside a neighborhood of the
attracting invariant manifold in a finite time. _

###  6.4 The inverse conjugation: proof of Corollary 2.18

Since ${\mathcal{H}}_{1}(\theta,\psi)=\theta+\theta^{2}h_{1}(\theta,\psi)$ and
$h_{1}\in{\mathscr{B}}_{\alpha_{*},1}({\Omega_{1}},\mathds{R})$ for a suitable
$\alpha_{*}>0$, there exists $\theta_{*}\leq\theta_{1}$ such that
$\partial_{\theta}{\mathcal{H}}_{1}(\theta,\psi)>1/2$ for
$|\theta|<\theta_{*}$. For every $(\theta,\psi)\in\Omega_{1}$ and every
$M\in\mathds{N}$, we have

${\mathcal{H}}_{1}(\theta,\psi)=\frac{{\mathcal{H}}_{1}({\mathscr{S}}_{1}^{M}(\theta,\psi))}{\kappa^{(M)}(\psi)}\,.$

Reasoning like in the derivation of (6.34) we see that there exists $M_{2}$
such that $|({\mathscr{S}}_{1}^{M_{2}}(\theta,\psi))_{\theta}|\leq\theta_{*}$.
Thus, using Lemma 2.9 we get, for all $(\theta,\psi)\in\Omega_{1}$,

$\partial_{\theta}{\mathcal{H}}_{1}(\theta,\psi)\geq\|\kappa\|_{\infty}^{-M_{2}}\inf_{|\theta|\leq\theta^{\star}}\partial_{\theta}{\mathcal{H}}_{1}(\theta,\psi)\left(\inf_{(\theta,\psi)\in\Omega_{1}}\partial_{\theta}G_{1}(\theta,\psi)\right)^{M_{2}}=:\tau_{1}>0\,,$
(6.42)

where we used Hypothesis 1. From the Inverse Function Theorem it follows
easily that there exist
$l_{1}\in{\mathscr{B}}_{0,1}({\mathscr{H}}_{1}(\Omega_{1},\mathds{R}))$ such
that

$({\mathscr{H}}_{1}^{-1})_{\eta}(\eta,\psi)=\eta+\eta^{2}l_{1}(\eta,\psi)\,.$

Finally, for any $\psi,\psi^{\prime}\in\mathds{T}^{2}$, we can write

${\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi^{\prime}),\psi^{\prime})-{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)=0,$

so that

$\frac{|{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi^{\prime}),\psi^{\prime})-{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi^{\prime})|}{|\psi^{\prime}-\psi|^{\alpha_{*}}}=\frac{|{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi^{\prime})-{\mathcal{H}}_{1}({\mathcal{H}}_{1}^{-1}(\eta,\psi),\psi)|}{|\psi^{\prime}-\psi|^{\alpha_{*}}}\,,$

which gives

$\left(\inf_{(\theta,\psi)\in\Omega_{1}}\partial_{\theta}{\mathcal{H}}_{1}(\theta,\psi)\right)|{\mathcal{H}}_{1}^{-1}|_{\alpha_{*}}\leq|{\mathcal{H}}_{1}|_{\alpha_{*}}\,,$

and hence
$|{\mathcal{H}}_{1}^{-1}|_{\alpha_{*}}\leq\tau_{1}^{-1}|{\mathcal{H}}_{1}|_{\alpha_{*}}$.
Therefore, the fact that
$h_{1}\in{\mathscr{B}}_{\alpha_{*},1}(\Omega_{1},\mathds{R})$ implies also
that
$l_{1}\in{\mathscr{B}}_{\alpha_{*},1}({\mathscr{H}}_{1}(\Omega_{1}),\mathds{R})$,