# Mean field coupled dynamical systems: Bifurcations and phase transitions

Wael Bahsoun  Department of Mathematical Sciences, Loughborough University,
Loughborough, Leicestershire, LE11 3TU, UK<EMAIL_ADDRESS>and
Carlangelo Liverani  Dipartimento di Matematica
II Università di Roma (Tor Vergata)
Via della Ricerca Scientifica, 00133 Roma, Italy<EMAIL_ADDRESS>
###### Abstract.

We develop a bifurcation theory for infinite dimensional systems satisfying
abstract hypotheses that are tailored for applications to mean field coupled
chaotic maps. Our abstract theory can be applied to many cases, from globally
coupled expanding maps to globally coupled Axiom A attractors. To illustrate
the range of applicability, we analyze an explicit example consisting of
globally coupled Anosov diffeomorphisms. For such an example, we classify all
the invariant measures as the coupling strength varies; we show which
invariant measures are _physical_ , and we prove that the existence of
multiple invariant physical measures is a purely infinite dimensional
phenomenon, i.e., the model exhibits phase transitions in the sense of
statistical mechanics.

The research of W. Bahsoun (WB) was supported by EPSRC grant EP/V053493/1. The
research of C. Liverani (CL) was supported by the PRIN Grants “Regular and
stochastic behavior in dynamical systems” (PRIN 2017S35EHN), “Stochastic
properties of dynamical systems” (PRIN 2022NTKXCX) and by the MIUR Excellence
Department Project Math@TOV awarded to the Department of Mathematics,
University of Rome Tor Vergata. CL acknowledges the membership of GNFM/INDAM.
CL thanks the Department of Mathematical Sciences of the University of
Loughborough for supporting his visit during the completion of this work. WB
thanks the Department of Mathematics at the University of Rome Tor Vergata for
its hospitality during the course of this work. WB and CL would like to thank
the hospitality of MATRIX at the University of Melbourne and the School of
Mathematics and Statistics at University of New South Wales where part of this
research was conducted. Our participation in the MATRIX programme was
supported by the Simons Foundation. C.L. thanks Sebastien Gouëzel for a
helpful discussion.

###### Contents

  1. 1 Introduction
  2. 2 The model and an abstract bifurcation theory
  3. 3 Regularity properties
    1. 3.1 Properties of the functions $H$ and $F$
  4. 4 Charcterization of invariant measures
    1. 4.1 Local existence of invariant measures
    2. 4.2 Global existence of invariant measures
  5. 5 Local stability and physical measures
  6. 6 Example
    1. 6.1 Phase transition
  7. A A quantitative Implicit Function Theorem

##  1\. Introduction

The study of large, possibly infinite-dimensional, coupled systems is of
growing importance in many fields, including mathematics, physics and biology,
see for example [6, 7, 20, 27] and references therein. A far-reaching class of
coupled systems is the _globally coupled_ one, also called _mean field_
systems, of interacting units in which each unit interacts with all other
units of the system. In the field of dynamical systems, such systems provide
an interesting alternative to locally coupled map lattices that were
investigated by many authors, starting with Kaneko [15], and were first put in
a rigorous setting by Bunimovich and Sinaĭ [9].

In physics, mean field systems are of paramount importance and have been
studied for a long time. They are described by the Vlasov equation (see [25]),
which is far from being completely understood as it may exhibit unexpected
phenomena, e.g. Landau damping [19].

Recently, Keller [18] investigated mean filed systems when the dynamics of
each unit is described by an expanding map, and introduced the idea that, in
the infinite size limit, a _nonlinear_ transfer operator can describe the
evolution of states of the underlying system. This plays a role simlar to the
Vlasov equation for gas models.

Following the work of Keller [18], a considerable amount of work has been done
on globally coupled systems where the dynamics of each unit is described by an
expanding map [4, 8, 23, 12],111The work of [12] contains a general framework
when the site dynamics admits exponential decay of correlations and also
applies to certain coupled random systems. and more recently by an Anosov map
[3] or by an intermittent map [1] (see also [26] for a recent review). Most of
the above results pertain to the case of _weak_ coupling strength where the
existence of a unique invariant measure with appropriate characteristics (e.g.
being _physical_) is proven. However, there are few results in the literature
in which it is proven that a globally coupled system admits several invariant
measures ([5, 22]). Indeed, in [5] an example of a globally coupled expanding
map where the infinite system admits exactly three absolutely continuous
invariant measures is discussed. Two of the invariant measures of the infinite
system are _stable_ ; i.e., physical in the strong sense of this paper, while
the third one is unstable. Moreover, it is shown that in the finite-
dimensional case, the system preserves exactly one absolutely continuous
invariant measure. However, the example is ad hoc and is explicitly
constructed to exhibit multiple invariant measures. A general theory to
explore all the invariant measures of given systems is missing.
Our goal is to start developing a general theory describing the invariant
measures and their characteristics for globally coupled systems when the
coupling strength is not necessarily weak, but the system still retains a
chaotic behavior. Since such a problem can be reduced to the study of the
fixed points of an infinite dimensional dynamical system described by a
nonlinear operator acting on an appropriate Banach space, it is natural to
wonder if some elementary facts of bifurcation theory can be extended to this
context.

In this work, we first develop a general theory to study bifurcations in
infinite dimensional systems satisfying some abstract hypotheses. These
hypotheses are not the standard ones under which the bifurcation theory of
infinite dimensional systems is studied, they are tailored for applications to
globally coupled chaotic maps.

To show that the general theory we develop applies to cases of interest, we
study a simple and natural example in which we consider globally coupled
Anosov diffeomorphisms. In this example, we show that in the weak coupling
regime, the system admits a unique physical measure, while for larger coupling
strength the system admits multiple physical measures (see Proposition 6.7 for
a precise statement). More importantly, we show that the latter aspect is only
an infinite-dimensional phenomenon; i.e., it appears only in the infinite-
dimensional limit (thermodynamical limits). Hence, we are in the presence of
phase transitions in the sense of statistical mechanics [21, 24], and the
bifurcation diagram can be viewed as a phase transition diagram.

It is worth noting that our abstract theory can be applied to other classes of
infinite dimensional systems, e.g. globally coupled expanding maps, general
Axiom A maps, and partially hyperbolic maps for which one has an adequate
understanding of the transfer operator (possibly [10]), but we believe that
our example suffices to illustrate the range of applicability of our abstract
results.

The paper is organized as follows: In section 2, we present an abstract
bifurcation theory adapted to our needs, then we discuss a general class of
globally coupled systems, and we state the main abstract results of our work
(Theorem 2.6, Theorem 2.8 and Theorem 2.12). In Section 3 we study the
differentiability properties of the nonlinear transfer operator. Section 4
contains the proof of the two first main theorems on the existence of the
invariant measures and their dependence on the coupling strength. Section 5
contains the proof of Theorem 2.12 and discusses the properties of the above
invariant measures: in particular, it provides a criterion for identifying
physically observable invariant measures. Section 6 discusses in detail an
example showing the presence of a large interval of coupling strength for
which there exist many physical invariant measures. It is important to note
that, contrary to most of the literature on the subject, we can use our
general theory to treat coupling strength well beyond the perturbative regime.
Moreover, we show that the existence of many invariant states is a purely
infinite dimensional phenomenon (section 6.1); that is, we are in the presence
of a phase transition. Finally, in Appendix A, we provide an explicit
quantitative version of the infinite-dimensional implicit function theorem
since we had trouble locating it in the literature. Indeed, such a theorem is
often stated in nonquantitative terms, while we need an estimate of the domain
in which the theorem applies.

##  2\. The model and an abstract bifurcation theory

We are interested in the following class of dynamical systems.
Let $T\in{\mathcal{C}}^{\infty}(\mathbb{T}^{d},\mathbb{T}^{d})$,222 Here, and
in the following, we use $\mathcal{C}^{\infty}$ to simplify notation, in fact
${\mathcal{C}}^{r}$, with e.g. $r\geq 6$, would suffice. Also, the theory can
be easily extended to maps on a compact manifold $M$, yet we decided to keep
things simple. and ${\mathbb{W}}_{0}=\\{h\in L^{1}({\mathbb{T}}^{d})\;:\;h\geq
0,\int h=1\\}$. For $h\in{\mathbb{W}}_{0}$, and $\nu\in{\mathbb{R}}$ let

$T_{\nu,h}(x)=T(x)+\nu\beta(x)\int\alpha(y)h(y)dy\quad\text{mod }1,$ (2.1)

where $\alpha\in C^{\infty}(\mathbb{T}^{d},{\mathbb{R}})$ and $\beta\in
C^{\infty}(\mathbb{T}^{d},{\mathbb{R}}^{d})$. We call the system in (2.1) a
mean field coupled system, or globally coupled map, of infinitely many
interacting particles with coupling strength $\nu$ and state $h$. See [3] for
an explanation of this language, physical motivations and for references.

######  Remark 2.1.

Here, for simplicity, we restrict ourselves to systems of type (2.1), but we
could consider an integral operator with a more general kernel. It is also
possible to study the case in which the mean-field influences a parameter of
the map, via a feedback function, in the spirit of [5].

Let ${\mathcal{L}}_{S}$ denote the transfer operator associated with a non-
singular map $S\in{\mathcal{C}}^{\infty}({\mathbb{T}}^{d},{\mathbb{T}}^{d})$;
more precisely, for $f\in L^{1}(\mathbb{T}^{d}),g\in
L^{\infty}(\mathbb{T}^{d})$

$\int_{\mathbb{T}^{d}}g\cdot{\mathcal{L}}_{S}fdm:=\int_{\mathbb{T}^{d}}g\circ
S\cdot fdm.$ (2.2)

Consequently
$\int_{\mathbb{T}^{d}}{\mathcal{L}}_{S}fdm=\int_{\mathbb{T}^{d}}fdm$. Let us
define

$\widetilde{\mathcal{L}}_{\nu}(h)={\mathcal{L}}_{T_{\nu,h}}h.$ (2.3)

Then $({\mathbb{W}}_{0},\widetilde{\mathcal{L}}_{\nu})$ is an infinite
dimensional, nonlinear, dynamical system. The fixed points of
$\widetilde{\mathcal{L}}_{\nu}$ are invariant states of the coupled system
(2.1). Our goal is to study the number and properties of solutions of the
equation

$\widetilde{\mathcal{L}}_{\nu}(h_{\nu})=h_{\nu},$ (2.4)

as $\nu$ varies.

To this end, studying the action of ${\mathcal{L}}_{T_{\nu,h}}$ on appropriate
Banach spaces is convenient, and often necessary since the invariant measure
may not belong to $L^{1}$. Since our aim is to develop a general theory, we
study (2.4) under some abstract assumptions on the existence of Banach spaces
with certain properties. Such hypotheses hold, e.g., for expanding maps and
Anosov maps. The latter, which is less obvious, will be discussed explicitly
in section 6 where a concrete example is analyzed in detail.

Let
${\mathcal{T}}\subset{\mathcal{C}}^{\bar{r}}({\mathbb{T}}^{d},{\mathbb{T}}^{d})$
be a set closed by compositions. Assume that there exists $\nu_{*}>0$ such
that, for all $\nu\in[0,\nu_{*}]$ and $h\in{\mathbb{W}}_{0}$,
$T_{\nu,h}\in{\mathcal{T}}$.
Let $\\{{\mathcal{B}}_{i}\\}_{i=0}^{3}$ be Banach spaces such that, for some
$\bar{r}\in{\mathbb{N}}$,333 The inclusion is meant in the sense that there
exists a one-to-one embedding. Also,
$\|h\|_{{\mathcal{B}}_{i}}\leq\|h\|_{{\mathcal{B}}_{i+1}}$. While the
embedding in $(C^{\bar{r}}(\mathbb{T}^{d}))^{\prime}$ if the natural extension
of the embedding of
$\iota:C^{\bar{r}}(\mathbb{T}^{d})\to((C^{\bar{r}}(\mathbb{T}^{d}))^{\prime}$
given by $\iota(h)(\phi)=\int h\phi$, for each $\phi\in
C^{\bar{r}}(\mathbb{T}^{d})$.

$\begin{split}&(C^{\bar{r}}(\mathbb{T}^{d}))^{\prime}\supset{\mathcal{B}}_{0}\supset{\mathcal{B}}_{1}\supset\cdots\supset{\mathcal{B}}_{3}\supset
C^{\bar{r}}(\mathbb{T}^{d})\\\
&(C^{\bar{r}}(\mathbb{T}^{d}))^{\prime}\supset{\mathcal{B}}_{3}^{\prime}\supset{\mathcal{B}}_{2}^{\prime}\supset\cdots\supset{\mathcal{B}}_{0}^{\prime}\supset
C^{\bar{r}}(\mathbb{T}^{d})\end{split}$ (2.5)

and such that $C^{\bar{r}}(\mathbb{T}^{d})$ is dense in each
${\mathcal{B}}_{i},{\mathcal{B}}_{i}^{\prime}$. For $K_{\star}>0$ large
enough, let444 By $h\geq 0$ we mean that for all
$\varphi\in{\mathcal{C}}^{\bar{r}}$, $\varphi\geq 0$, $\int h\varphi\geq 0$.

$\begin{split}&{\mathbb{W}}=\left\\{h\in{\mathcal{B}}_{1}:\|h\|_{{\mathcal{B}}_{1}}\leq
K_{\star},h\geq 0,\int h=1\right\\}\\\
&{\mathbb{V}}=\left\\{f\in{\mathcal{B}}_{1}:\,\int f=0\right\\}.\end{split}$

For all $S\in{\mathcal{T}}$, equation (2.2) defines naturally the operator
${\mathcal{L}}_{S}$ as an element of
$L((C^{\bar{r}}(\mathbb{T}^{d}))^{\prime},(C^{\bar{r}}(\mathbb{T}^{d}))^{\prime})$.
We can then restrict ${\mathcal{L}}_{S}$ to each ${\mathcal{B}}_{i}$. To ease
notation, we will call such restrictions ${\mathcal{L}}_{S}$ as well. Note
that, if ${\mathcal{L}}_{S}\in L({\mathcal{B}}_{1},{\mathcal{B}}_{1})$, then
equation (2.2) implies ${\mathcal{L}}_{S}{\mathbb{V}}\subset{\mathbb{V}}$.
We assume that there exists $K,C_{*}\geq 1$, $\bar{c},\hat{c}>0$, and
$\vartheta\in(0,1)$ such that,

  1. (A1)

For each $S\in{\mathcal{T}}$, ${\mathcal{L}}_{S}\in
L({\mathcal{B}}_{i},{\mathcal{B}}_{i})$ for all $i\in\\{0,\dots,3\\}$.

  2. (A2)

For each $k\in\\{1,\dots,d\\}$, and $i\in\\{0,\dots,2\\}$,
$\partial_{x_{k}}\in L({\mathcal{B}}_{i+1},{\mathcal{B}}_{i}\\}$.

  3. (A3)

For any $a\in{\mathcal{C}}^{\bar{c}i+\hat{c}}$,
$\|ah\|_{{\mathcal{B}}_{i}}\leq
C_{\\#}\|a\|_{{\mathcal{C}}^{\bar{c}i+\hat{c}}}\|h\|_{{\mathcal{B}}_{i}}$,
$\left|\int h\right|\leq C_{*}\|h\|_{{\mathcal{B}}_{0}}$.

  4. (A4)

The unit ball of ${\mathcal{B}}_{i+1}$ is relatively compact in
${\mathcal{B}}_{i}$ for $i\in\\{0,1,2\\}$.

  5. (A5)

For all $n\in{\mathbb{N}}$, $\\{S_{j}\\}_{j=1}^{n}\subset{\mathcal{T}}$ and
$\phi\in{\mathcal{B}}_{i}$, $i\in\\{1,\dots,3\\}$,

$\begin{split}&\|{\mathcal{L}}_{S_{n}}\cdots{\mathcal{L}}_{S_{1}}\phi\|_{{\mathcal{B}}_{0}}\leq
C_{*}\|\phi\|_{{\mathcal{B}}_{0}}\\\
&\|{\mathcal{L}}_{S_{n}}\cdots{\mathcal{L}}_{S_{1}}\phi\|_{{\mathcal{B}}_{i}}\leq
C_{*}\vartheta^{n}\|\phi\|_{{\mathcal{B}}_{i}}+K\|\phi\|_{{\mathcal{B}}_{i-1}}.\\\
\end{split}$

  6. (A6)

The exists $\bar{n}\in{\mathbb{N}}$ such that, for all
$\\{S_{j}\\}_{j=1}^{n}\subset{\mathcal{T}}$ , $n\geq\bar{n}$,
$i\in\\{1,\dots,3\\}$,

$\|{\mathcal{L}}_{S_{n}}\cdots{\mathcal{L}}_{S_{1}}|_{{\mathbb{V}}}\|_{{\mathcal{B}}_{i}}<\vartheta^{n}.$

In particular, the above implies that, for $K_{\star}$ large enough
${\mathcal{L}}_{S}{\mathbb{W}}\subset{\mathbb{W}}$.

######  Remark 2.2.

It is worth noting that (A4) and (A5) imply that the operators
${\mathcal{L}}_{S}$ are quasi-compact. Assumption (A6) implies that
${\mathcal{L}}_{S}$ has a spectral gap on ${\mathcal{B}}_{i}$, $i=1,2,3$, see
Lemma 2.3. Note that the existence of a spectral gap is weaker than Assumption
(A6) since the compositions can comprise very different operators.

The first implication of the above assumptions is the following.

######  Lemma 2.3.

Under assumptions (A5) and (A6), for each
$\\{S_{j}\\}_{j=1}^{\bar{n}}\subset{\mathcal{T}}$ we have
${\mathcal{L}}_{S_{\bar{n}}}\cdots{\mathcal{L}}_{S_{1}}{\mathbb{W}}\subset{\mathbb{W}}$.
In additon, for each $S\in{\mathcal{T}}$,
$\sigma_{{\mathcal{B}}_{1}}({\mathcal{L}}_{S})\subset\\{1\\}\cup\\{z\in{\mathbb{C}}\;:\;|z|<\vartheta\\}$
and $1$ is a simple eigenvalue. Finally, if $h\in L^{1}$, $\int h=1$, is such
that ${\mathcal{L}}_{S}h=h$, then $h\in{\mathbb{W}}$.

###### Proof.

Note that, for each $h\in{\mathbb{W}}$ and
$\\{S_{j}\\}_{j=1}^{\bar{n}}\subset{\mathcal{T}}$, assumption (A5) implies,
for each $n\in{\mathbb{N}}$,

$\begin{split}&\|{\mathcal{L}}_{S_{n}}\cdots{\mathcal{L}}_{S_{1}}1\|_{{\mathcal{B}}_{1}}\leq
C_{*}\|1\|_{{\mathcal{B}}_{1}}+K\|1\|_{{\mathcal{B}}_{0}}=:K_{1}\\\
&\|{\mathcal{L}}_{S_{n}}\cdots{\mathcal{L}}_{S_{1}}h\|_{{\mathcal{B}}_{1}}\leq
C_{*}\|h\|_{{\mathcal{B}}_{1}}+K\|h\|_{{\mathcal{B}}_{0}}\leq(C_{*}+K)K_{\star}.\end{split}$
(2.6)

Then Assumption (A6) implies

$\begin{split}\|{\mathcal{L}}_{S_{\bar{n}}}\cdots{\mathcal{L}}_{S_{1}}h\|_{{\mathcal{B}}_{1}}&\leq\|{\mathcal{L}}_{S_{\bar{n}}}\cdots{\mathcal{L}}_{S_{1}}(h-1)\|_{{\mathcal{B}}_{1}}+K_{1}\\\
&\leq\vartheta^{\bar{n}}\|h-1\|_{{\mathcal{B}}_{1}}+K_{1}\leq\vartheta^{\bar{n}}\|h\|_{{\mathcal{B}}_{1}}+2K_{1}\\\
&\leq K_{\star}\vartheta+2K_{1}\leq K_{\star}\end{split}$ (2.7)

provided $K_{\star}$ is large enough. Hence,
${\mathcal{L}}_{S_{\bar{n}}}\cdots{\mathcal{L}}_{S_{1}}{\mathbb{W}}\subset{\mathbb{W}}$.
To prove the second statement, note that
$\|{\mathcal{L}}_{S}^{\bar{n}}|_{{\mathbb{V}}}\|_{{\mathcal{B}}_{1}}<\vartheta^{\bar{n}}$,
hence
$\sigma({\mathcal{L}}_{S}|_{{\mathbb{V}}})\subset\\{z\in{\mathbb{C}}\;:\;|z|<\vartheta\\}$.
The lemma follows since ${\mathbb{V}}$ has codimension one. To conclude the
lemma assume that $h\in L^{1}$, $\int h=1$, is such that
${\mathcal{L}}_{S}h=h$. Then, for all $\varepsilon>0$, by density there exist
$h_{\varepsilon}\in{\mathcal{C}}^{\infty}$, $\int h_{\varepsilon}=1$, such
that $\|h-h_{\varepsilon}\|_{L^{1}}\leq\varepsilon$, and, by the definition
(2.2),

$\|{\mathcal{L}}_{S}^{n}h_{\varepsilon}-{\mathcal{L}}_{S}^{n}h\|_{L^{1}}\leq\varepsilon.$

Since ${\mathcal{L}}_{S}$ has 1 as a simple eigenvalue, whose eigenfunction we
denote by $h_{S}\in{\mathcal{B}}_{1}$, and a spectral gap, for each
$\varphi\in{\mathcal{C}}^{0}$ we have

$\int\varphi
h=\lim_{n\to\infty}\int\varphi{\mathcal{L}}_{S}^{n}h=\lim_{n\to\infty}\int\varphi{\mathcal{L}}_{S}^{n}h_{\varepsilon}+{\mathcal{O}}(\varepsilon)=\int\varphi
h_{S}+{\mathcal{O}}(\varepsilon).$

Since $\varepsilon$ is arbitrary by (2.5) it follows $h=h_{S}$. In addition,
arguing as in (2.7), implies

$\|h_{S}\|_{{\mathcal{B}}_{1}}=\lim_{k\to\infty}\|{\mathcal{L}}^{k\bar{n}}_{S}h_{S}\|_{{\mathcal{B}}_{1}}\leq\lim_{k\to\infty}\vartheta^{k\bar{n}}\|h_{S}\|_{{\mathcal{B}}_{1}}+2K_{1}\leq
K_{\star},$

so $h\in{\mathbb{W}}$. ∎

Another simple consequence of the above assumptions is the existence of
periodic orbits.

######  Theorem 2.4.

Under our assumptions, for each $\nu\in[0,\nu_{*}]$,
$\widetilde{\mathcal{L}}_{\nu}$ has at least one periodic orbit in
${\mathcal{B}}_{1}$.

###### Proof.

By Lemma 2.3 we have
$\widetilde{\mathcal{L}}_{\nu}^{\bar{n}}({\mathbb{W}})\subset{\mathbb{W}}$.
Next, note that if $h,h_{1}\in{\mathbb{W}}$ are close enough, Assumptions
(A3), (A5), and Lemma 3.1 imply

$\begin{split}\|\widetilde{\mathcal{L}}_{\nu}(h)-\widetilde{\mathcal{L}}_{\nu}(h_{1})\|_{{\mathcal{B}}_{0}}&\leq\|{\mathcal{L}}_{T_{\nu,h}}(h)-{\mathcal{L}}_{T_{\nu,h_{1}}}(h)\|_{{\mathcal{B}}_{0}}+\|{\mathcal{L}}_{T_{\nu,h_{1}}}(h-h_{1})\|_{{\mathcal{B}}_{0}}\\\
&\leq(C_{*}+\bar{K}\|h\|_{{\mathcal{B}}_{1}})\|h-h_{1}\|_{{\mathcal{B}}_{0}}\leq(C_{*}+\bar{K}K_{\star})\|h-h_{1}\|_{{\mathcal{B}}_{0}},\end{split}$

for some constant $\bar{K}>0$. That is $\widetilde{\mathcal{L}}_{\nu}$ is
continuous on ${\mathbb{W}}$ in the ${\mathcal{B}}_{0}$ topology. In addition,
by Assumption (A4), ${\mathbb{W}}$ is compact in ${\mathcal{B}}_{0}$, and
clearly also convex. Then, by the Schauder fixed point Theorem, there exists
at least an $h_{*}\in{\mathbb{W}}$ such that
$\widetilde{\mathcal{L}}_{\nu}^{\bar{n}}h_{*}=h_{*}$.

Let $h_{*,k}:=\widetilde{\mathcal{L}}_{\nu}^{k}h_{*}$ and
$T_{k}=T_{\nu,h_{*,k-1}}\circ\dots\circ T_{\nu,h_{*,0}}$. Then
$h_{*}={\mathcal{L}}_{T_{\bar{n}}}h_{*}$. Since $T_{k}\in{\mathcal{T}}$ by
assumption, Assumption (6) implies that there exists a unique
$h_{0}\in{\mathcal{B}}_{1}$ such that
${\mathcal{L}}_{T_{\bar{n}}}h_{0}=h_{0}$. Hence, it must be
$h_{*}=h_{0}\in{\mathcal{B}}_{1}$. The wanted periodic orbit is then
$\\{\widetilde{\mathcal{L}}_{\nu}^{k}h_{0}\\}_{k=0}^{\bar{n}-1}$. ∎

######  Remark 2.5.

Notice that if $\bar{n}=1$, then the argument in the proof of Theorem 2.4
implies the existence of a fixed point for $\widetilde{\mathcal{L}}_{\nu}$ in
${\mathcal{B}}_{1}$.

Next, we want to explore not periodic orbits, but rather fixed points of
$\widetilde{\mathcal{L}}_{\nu}$. In this direction, there are already many
results on the existence of a unique fixed point of
$\widetilde{\mathcal{L}}_{\nu}$ for small $\nu$; see [12] for general results
and references. On the contrary, here we are interested in the non
perturbative regime, i.e. large $\nu$.

Lemma 2.3 implies that ${\mathcal{L}}_{T_{\nu,h}}$ has a unique invariant
probability measure which belongs to ${\mathbb{W}}$ (the SRB or physical
measure). Let us call it $H(\nu,h)$. We will prove further properties of
$H(\nu,h)$ in Lemma 3.4 below.

Let $\Omega:=[0,\nu_{*}]\times{\mathbb{W}}$ equipped with the topology of
${\mathbb{R}}\times{\mathcal{B}}_{0}$. Note that $\Omega$ is then a compact
set. Our basic idea is to study solutions of the following implicit equation

$F=0,$

where $F:\Omega\to{\mathbb{V}}$ is defined by

$F(\nu,h)=h-H(\nu,h).$ (2.8)

Note that $F(\nu,h)=0$ implies

$h=H(\nu,h)={\mathcal{L}}_{T_{\nu,h}}H(\nu,h)={\mathcal{L}}_{T_{\nu,h}}h=\widetilde{\mathcal{L}}_{\nu}(h)$

that is $h$ is a fixed point of $\widetilde{\mathcal{L}}_{\nu}$. Moreover, if
$\widetilde{\mathcal{L}}_{\nu}h=h$, assumption (A6) implies $h=H(\nu,h)$. We
will see in Lemma 3.6 that, under our assumptions,
$F\in{\mathcal{C}}^{2}(\Omega,{\mathcal{B}}_{1})$. For convenience, we define

$\Theta(\nu,h)=-({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\operatorname{div}{{\mathcal{L}}_{T_{\nu,h}}}\beta
H(\nu,h),$ (2.9)

which will appear in the expression of the Fréchet derivative, $D_{h}F$, of
$F$. We are now ready to state our main results. First we have a local result.

######  Theorem 2.6 (Local).

Suppose that there exists $\nu_{0}\in[0,\nu_{*})$ and $h_{0}\in{\mathbb{W}}$
such that

$F(\nu_{0},h_{0})=0,$ (2.10)

and either $D_{h_{0}}F$ is invertible or

$\displaystyle\operatorname{Range}(D_{h_{0}}F)(\nu_{0},\cdot)\oplus
D_{\nu}F(\nu_{0},h_{0})={\mathbb{V}};$ (H1) $\displaystyle\int\alpha
D^{2}_{h}F|_{\nu_{0},h_{0}}(\Theta(\nu_{0},h_{0}),\Theta(\nu_{0},h_{0}))\neq
0;$ (H2) $\displaystyle\int\alpha h_{0}\neq 0.$ (H3)

Then there exists $\delta>0$ and a differentiable map (in the
${\mathcal{B}}_{1}$ topology)
$(G,{\mathbbm{h}})=:(-\delta,\delta)\to[0,\nu_{*}]\times{\mathbb{W}}$,
$(G(0),{\mathbbm{h}}(0))=(\nu_{0},h_{0})$, such that all the solutions of the
equation $F(\nu,h)=0$ in
$\\{(\nu,h)\in{\mathbb{R}}\times{\mathbb{W}}\;:\;|\nu-\nu_{0}|+\|h-h_{0}\|_{{\mathcal{B}}_{1}}\leq\delta\\}$
belong to the set ${\mathbbm{h}}(-\delta,\delta)$. In addition, if $\nu_{0}>0$
and $D_{h}F(\nu_{0},h_{0})$ is not invertible, then there exists
$\nu\in(0,\nu_{*})$ for which the equation $F(\nu,h)=0$ has a least two
solutions.

The proof of the above theorem is contained in Section 4.1.

######  Remark 2.7.

Note that Theorem 2.6 implies linear response; i.e., differentiability in the
${\mathcal{B}}_{1}$ topology, with respect to the parameter $\nu$, of the
invariant measures.

Next, we establish a global result, which characterizes all the fixed points
of $\widetilde{\mathcal{L}}_{\nu}$ (the invariant measures).

######  Theorem 2.8 (Global).

Suppose that for each $\nu\in[0,\nu_{*}]$, $h\in{\mathbb{W}}$ such that

$F(\nu,h)=0,$ (2.11)

either $D_{h}F$ is invertible or

$\displaystyle\operatorname{Range}(D_{h}F)(\nu,\cdot)\oplus
D_{\nu}F(\nu,h)={\mathbb{V}};$ $\displaystyle\int\alpha
D^{2}_{h}F|_{\nu,h}(\Theta(\nu,h),\Theta(\nu,h))\neq 0;$
$\displaystyle\int\alpha h\neq 0.$

Then there exist finitely many intervals $I_{k}\subset{\mathbb{R}}$ and
differentiable curves
$(G_{k},{\mathbbm{h}}_{k}):I_{k}\to[0,\nu^{*}]\times{\mathcal{B}}_{1}$ such
that all the ${\mathcal{B}}_{1}$ fixed points of
$\widetilde{\mathcal{L}}_{\nu}$ belong to
$\cup_{k}{\mathbbm{h}}_{k}(I_{k})\subset{\mathcal{B}}_{3}$. More precisely
$\widetilde{\mathcal{L}}_{G_{k}(y)}{\mathbbm{h}}_{k}(y)={\mathbbm{h}}_{k}(y)$
for all $y\in I_{k}$. Moreover, $\sharp\\{G_{k}^{-1}(\nu)\\}<\infty$ for all
$\nu\in[0,\nu_{*}]$. Finally, if for some $h\in{\mathbb{W}}$,
$\nu\in(0,\nu_{*})$, $F(\nu,h)=0$ and $D_{h}F$ is not invertible, then there
exists $\nu_{1}\in(0,\nu_{*})$ such that
$\sharp\\{\bigcup_{k}G_{k}^{-1}(\nu_{1})\\}\geq 3$ (that is, there are at
least three invariant measures for $\widetilde{\mathcal{L}}_{\nu_{1}}$).

The proof of the above Theorem can be found in section 4.2.
Figures 1 and 2 below provide two pictorial illustrations for possible
bifurcation diagrams in Theorem 2.8.

Figure 1. A possible bifurcation diagram of $h_{\nu}$ in Theorem 2.8 Figure 2.
Another possible scenario for the bifurcation diagram of $h_{\nu}$ in Theorem
2.8

We conclude this section with a discussion and a theorem concerning the
physical relevance of the measures appearing in Theorems 2.6 and 2.8. In [3],
we defined a physical measure as one that can be realized starting from an
absolutely continuous initial measure. However, such a definition does not
capture the intuitive meaning of physical measure completely, which is that
such measures describe the asymptotic behavior of a large class of initial
conditions, and small changes in the initial conditions do not change the
asymptotic measure. In the case at hand, the initial condition is a measure.
Hence, it is natural to consider an absolutely continuous initial measure, in
analogy with the finite-dimensional case, and this was the motivation of the
definition in [3]. Yet, it is also natural to ask that a small change in the
initial measure does not change the asymptotic behavior of the system.
Hence, we will call the measures that satisfy the condition in [3] visible,
and we introduce the following new definition to characterize physical
measures that include such a natural stability property. This is inspired, but
different, from the idea of stability used in [5, 22]. Let $d_{w}$ be a metric
in the space of measures that metrizes the weak topology.

######  Definition 2.9.

An invariant measure $h$ of $T_{\nu,h}$, that is $(T_{\nu,h})_{*}(h)=h$, is
called a visible measure if there exists $g\in{\mathcal{C}}^{\infty}$ such
that

$\lim_{n\to\infty}d_{w}(\widetilde{\mathcal{L}}_{\nu}^{n}(g),h)=0.$

######  Definition 2.10.

An invariant measure $h$ of $T_{\nu,h}$, that is $(T_{\nu,h})_{*}(h)=h$, is
called a physical measure if there exists an open set
${\mathcal{A}}\subset{\mathcal{C}}^{\infty}({\mathbb{T}}^{d},{\mathbb{R}}_{+})\setminus\\{0\\}$
such that, for any sequences
$m_{n}(\mu)=(T_{\nu,m_{n-1}(\mu)})_{*}m_{n-1}(\mu)$, $n\in{\mathbb{N}}$,
$m_{0}(\mu):=\mu\in{\mathcal{M}}_{{\mathcal{A}}}$, where

${\mathcal{M}}_{{\mathcal{A}}}=\left\\{\mu:\;\frac{d\mu}{d\operatorname{Leb}}=\rho,\;\rho=\frac{g}{\int
g}\;:\;g\in{\mathcal{A}}\right\\},$

we have

$\lim_{n\to\infty}\sup_{\mu\in{\mathcal{M}}_{{\mathcal{A}}}}d_{w}(m_{n}(\mu),h)=0.$

######  Remark 2.11.

It is worth noting that the unique physical measures identified in [3] are
indeed physical also in this stronger sense, see in [3, Lemma 3.13].

For $z\not\in\sigma({\mathcal{L}}_{T_{\nu,h}})\setminus\\{1\\}$, define the
functions

$\begin{split}\Theta(z)&:=-\nu(z{\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta
h\\\ \Xi(z)&=\int\alpha\Theta(z).\end{split}$ (2.12)

######  Theorem 2.12.

For $\nu\in[0,\nu_{*}]$, if $h$ is a visible invariant measure, then
$h\in{\mathcal{B}}_{2}$. Moreover, calling ${\mathcal{D}}_{h}$ be the Fréchet
derivative of $\widetilde{\mathcal{L}}_{\nu}$ at $h$,

${\mathcal{D}}_{h}(\phi)={\mathcal{L}}_{T_{\nu,h}}\phi-\nu\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta
h\int\alpha\phi.$

Finally, if
$\sigma({\mathcal{D}}_{h})\setminus\\{1\\}\subset\\{z\in{\mathbb{C}}\;:\;|z|<\kappa\\}$
for some $\kappa\in(0,1)$, and $\Xi(1)\neq 1$, then $h$ is a physical measure.
On the contrary, if there exists $z_{1}\in\sigma({\mathcal{D}}_{h})$,
$|z_{1}|>1$, then $h$ is not a physical measure.

The proof of the above theorem can be found at the end of section 5.

##  3\. Regularity properties

Our first task is to explore the regularity of maps ${\mathcal{L}}_{S(u)}$,
where $S$ is a map in
${\mathcal{C}}^{2}({\mathbb{R}},{\mathcal{C}}^{\bar{r}}({\mathbb{T}}^{2},{\mathbb{T}}^{2}))$.
For each compact interval $U\subset{\mathbb{R}}$ and
$S\in{\mathcal{C}}^{2}(U,{\mathcal{C}}^{\bar{r}}({\mathbb{T}}^{d},{\mathbb{T}}^{d}))$,
we set, for all
$\psi\in{\mathcal{C}}^{\infty}({\mathbb{T}}^{2},{\mathbb{T}}^{2})$ and $u\in
U$

${\mathcal{P}}_{u}\psi=-\operatorname{div}{\mathcal{L}}_{S(u)}(S^{\prime}(u)\psi).$
(3.1)

######  Lemma 3.1.

There exists $C_{*}>0$ such that, provided $\bar{r}$ is large enough, for each
compact interval $U\subset{\mathbb{R}}$ and
$S\in{\mathcal{C}}^{2}(U,{\mathcal{C}}^{\bar{r}}({\mathbb{T}}^{d},{\mathbb{T}}^{d}))$,
we have, for all $\psi\in{\mathcal{B}}_{i+1}$ and $s\in{\mathbb{R}}$

$\begin{split}&\|{\mathcal{P}}_{u}\psi\|_{{\mathcal{B}}_{i}}\leq
C_{*}\|S\|_{{\mathcal{C}}^{\bar{r}}}\|\psi\|_{{\mathcal{B}}_{i+1}},\quad
i\in\\{0,1,2\\}\\\
&\|({\mathcal{L}}_{S(u+s)}-{\mathcal{L}}_{S(u)})\psi\|_{{\mathcal{B}}_{i}}\leq
C_{*}s\|S\|_{{\mathcal{C}}^{\bar{r}}}\|\psi\|_{{\mathcal{B}}_{i+1}},\quad
i\in\\{0,1,2\\}\\\
&\|({\mathcal{L}}_{S(u+s)}-{\mathcal{L}}_{S(u)}-s{\mathcal{P}}_{u})\psi\|_{{\mathcal{B}}_{i}}\leq
C_{*}s^{2}\|S\|_{{\mathcal{C}}^{\bar{r}}}^{2}\|\psi\|_{{\mathcal{B}}_{i+2}},\quad
i\in\\{0,1\\}.\end{split}$

###### Proof.

The first inequality follows immediately from Assumption (A2), (A3) and (A5).
Next, we study the continuity of
${\mathcal{L}}_{S(\cdot)}:{\mathbb{W}}\subset{\mathcal{B}}_{0}\to\
L({\mathcal{B}}_{i},{\mathcal{B}}_{i})$. For
$\varphi,\psi\in{\mathcal{C}}^{\infty}$, we can compute

$\begin{split}&\int\varphi\left({\mathcal{L}}_{S(u+s)}-{\mathcal{L}}_{S(u)}\right)\psi=\int\varphi\circ
S(u+s)\psi-\int\varphi\circ S(u)\psi\\\
&=\int_{u}^{u+s}\int\langle(\nabla\varphi)\circ
S(\tau),S^{\prime}(\tau)\rangle\psi d\tau\\\
&=\int_{u}^{u+s}\int\langle(\nabla\varphi),{\mathcal{L}}_{S(\tau)}S^{\prime}(\tau)\psi\rangle\\\
&=\int_{u}^{u+s}\int\varphi\\{-\operatorname{div}{\mathcal{L}}_{S(\tau)}S^{\prime}(\tau)\psi\\}ds.\end{split}$
(3.2)

It follows that, for each $i\in\\{0,1,2\\}$, we have, recalling Assumptions
(A2), (A3) and (A5),

$\begin{split}&\left|\int\varphi\left({\mathcal{L}}_{S(u+s)}-{\mathcal{L}}_{S(u)}\right)\psi\right|\leq
s\|\varphi\|_{{\mathcal{B}}_{i}^{\prime}}\sup_{\tau\in[u,u+s]}\left\|\operatorname{div}{\mathcal{L}}_{S(\tau)}S^{\prime}(\tau)\psi\right\|_{{\mathcal{B}}_{i}}\\\
&\leq
s\,C_{*}\|\varphi\|_{{\mathcal{B}}_{i}^{\prime}}\sup_{\tau\in[u,u+s]}\left\|{\mathcal{L}}_{S(\tau)}S^{\prime}(\tau)\psi\right\|_{{\mathcal{B}}_{i+1}}\\\
&\leq
s\,C_{*}\|\varphi\|_{{\mathcal{B}}_{i}^{\prime}}\|\psi\|_{{\mathcal{B}}_{i+1}}\|S^{\prime}\|_{{\mathcal{C}}^{\bar{c}(i+1)+\hat{c}}}\end{split}$

where ${\mathcal{B}}_{i}^{\prime}$ is the dual space of ${\mathcal{B}}_{i}$.
Taking the sup on
$\varphi\in{\mathcal{B}}_{1}^{\prime}\cap{\mathcal{C}}^{\infty}$, which is
dense in ${\mathcal{B}}_{i}^{\prime}$ by (2.5), with
$\|\varphi\|_{{\mathcal{B}}_{i}^{\prime}}\leq 1$, yields

$\left\|{\mathcal{L}}_{S(u+s)}\psi-{\mathcal{L}}_{S(u)}\psi\right\|_{{\mathcal{B}}_{i}}\leq
s\,C_{*}\|\psi\|_{{\mathcal{B}}_{i+1}}\|S\|_{{\mathcal{C}}^{\bar{r}}}.$

We can continue (3.2) and write

$\begin{split}&\int\varphi\left({\mathcal{L}}_{S(u+s)}\psi-{\mathcal{L}}_{S(u)}\psi+s\operatorname{div}{\mathcal{L}}_{S(u)}S^{\prime}(u)\psi\right)=\\\
&=\int_{u}^{u+s}\int\langle(\nabla\varphi)\circ
S(\tau),S^{\prime}(\tau)\rangle\psi-\langle(\nabla\varphi)\circ
S(u),S^{\prime}(u)\rangle\psi\\}d\tau\\\
&=\int_{u}^{u+s}\hskip-12.0ptd\tau\int_{u}^{\tau}\hskip-6.0ptd\tau^{\prime}\int\langle(\nabla\varphi)\circ
S(\tau),S^{\prime\prime}(\tau^{\prime})\rangle\psi+D^{2}_{S(\tau^{\prime})}\varphi(S^{\prime}(\tau^{\prime}),S^{\prime}(\tau))\psi\\\
&=-\sum_{i=1}^{d}\int_{u}^{u+s}\hskip-12.0ptd\tau\int_{u}^{\tau}\hskip-6.0ptd\tau^{\prime}\int\varphi\partial_{x_{i}}{\mathcal{L}}_{S(\tau)}(S_{i}^{\prime\prime}(\tau^{\prime})\psi)\\\
&\phantom{=}+\sum_{i,j=1}^{d}\int_{u}^{u+s}\hskip-12.0ptd\tau\int_{u}^{\tau}\hskip-6.0ptd\tau^{\prime}\int\varphi\partial_{x_{i}}\partial_{x_{j}}{\mathcal{L}}_{S(\tau^{\prime})}(S_{i}^{\prime}(\tau^{\prime})S^{\prime}_{j}(\tau)\psi),\end{split}$

which implies, recalling Assumptions (A2), (A3) and (A5),

$\begin{split}\left|\int\varphi\left({\mathcal{L}}_{S(u+s)}\psi-{\mathcal{L}}_{S(u)}\psi-s{\mathcal{P}}_{u}\psi\right)\right|\leq&C_{*}s^{2}\|\varphi\|_{{\mathcal{B}}_{i}^{\prime}}\Big{[}\|S^{\prime\prime}\|_{{\mathcal{C}}^{\bar{\alpha}(i+1)+\bar{\beta}}}\|\psi\|_{{\mathcal{B}}_{i+1}}\\\
&+\|S^{\prime}\|_{{\mathcal{C}}^{\bar{c}(i+2)+\hat{c}}}^{2}\|\psi\|_{{\mathcal{B}}_{i+2}}\Big{]}.\end{split}$

Taking the sup on
$\varphi\in{\mathcal{B}}_{i}^{\prime}\cap{\mathcal{C}}^{\infty}$, with
$\|\varphi\|_{{\mathcal{B}}_{i}^{\prime}}$, yields, for $\bar{r}$ large
enough,

$\left\|{\mathcal{L}}_{S(u+s)}\psi-{\mathcal{L}}_{S(u)}\psi-s{\mathcal{P}}_{u}\psi\right\|_{{\mathcal{B}}_{i}}\leq
C_{*}s^{2}\|S\|_{{\mathcal{C}}^{\bar{r}}}^{2}\|\psi\|_{{\mathcal{B}}_{i+2}}.$

∎

######  Remark 3.2.

Note that the third equation in the statement of Lemma 3.1 implies that

$\frac{d}{du}{\mathcal{L}}_{S(u)}={\mathcal{P}}_{u}$ (3.3)

if we interpret ${\mathcal{L}}_{S(u)}$ as a function from ${\mathbb{R}}$ to
$L({\mathcal{B}}_{i+2},{\mathcal{B}}_{i})$, $i\in\\{0,1\\}$.

We can now use the above Lemma to investigate the regularity of the operator
${\mathcal{L}}_{T_{\nu,h}}$ seen as a map from $\Omega$ to
$L({\mathcal{B}}_{i+2},{\mathcal{B}}_{i})$.

######  Lemma 3.3.

${\mathcal{L}}_{T_{\cdot,\cdot}}\\!\in{\mathcal{C}}^{1}(\Omega,L({\mathcal{B}}_{i+2},{\mathcal{B}}_{i}))$,
$i\in\\{0,1\\}$, so it is Fréchect Differentiable. The derivative given by,
for each $h_{1}\in{\mathbb{V}}$,

$\left[\left(\frac{\partial}{\partial
h}{\mathcal{L}}_{T_{\nu,h}}\right)\\!(h_{1})\right]\psi=\nu\\{-\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta\psi\\}\int\alpha
h_{1}.$ (3.4)

$\left(\frac{\partial}{\partial\nu}{\mathcal{L}}_{T_{\nu,h}}\right)\psi=-\left(\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta\psi\right)\int\alpha
h.$ (3.5)

###### Proof.

For each $h_{0}\in{\mathbb{W}}$, and $h_{1}\in{\mathbb{V}}$ we can write
$S(u)=T_{\nu,h_{0}+uh_{1}}$, $\|h_{1}\|_{{\mathcal{B}}_{0}}=1$. Then, by
(2.1),

$S^{\prime}(u)=\nu\beta\int\alpha h_{1}.$

Then we apply Lemma 3.1 to write

$\frac{d}{du}{\mathcal{L}}_{S(u)}\psi=-\nu\left(\operatorname{div}{\mathcal{L}}_{T_{\nu,h_{0}+uh_{1}}}\beta\psi\right)\int\alpha
h_{1}.$

Note that the above is the Gâteaux derivative of ${\mathcal{L}}_{T_{\nu,h}}$
with respect to $h$ in the direction $h_{1}$.

On the other hand, if we set $S(u)=T_{u,h_{0}}$, we have

$S^{\prime}(u)=\beta\int\alpha h_{0}.$

Hence, Lemma 3.1 implies

$\frac{d}{du}{\mathcal{L}}_{S(u)}\psi=-\left(\operatorname{div}{\mathcal{L}}_{T_{\nu,h_{0}+uh_{1}}}\beta\psi\right)\int\alpha
h_{0}.$

Then the Gâteaux derivatives $\frac{\partial}{\partial
h}{\mathcal{L}}_{T_{\nu,h}}$ and
$\frac{\partial}{\partial_{\nu}}{\mathcal{L}}_{T_{\nu,h}}$, by assumptions
(A2) and the second of Lemma 3.1, are continuous functions from $\Omega$ to
$L({\mathcal{B}}_{i+2},{\mathcal{B}}_{i})$, $i\in\\{0,1\\}$. Hence, by [11,
Proposition 3.2.15], ${\mathcal{L}}_{T_{\cdot,\cdot}}:\Omega\to
L({\mathcal{B}}_{i+2},{\mathcal{B}}_{i})$ is Fréchect Differentiable with
derivative given by (3.4) and (3.5). ∎

###  3.1. Properties of the functions $H$ and $F$

Our next goal is the study of the regularity of $F$, defined in (2.8), hence
we have to start studying

$H:\Omega\to{\mathcal{B}}_{i},$

where $\Omega:=[0,\nu_{*}]\times{\mathbb{W}}$ is equipped with the topology of
${\mathbb{R}}\times{\mathcal{B}}_{0}$. Note that, by Assumption (A4), $\Omega$
is then a compact set. Assumption (A6) implies

${\mathcal{L}}_{T_{\nu,h}}=\Pi_{T_{\nu,h}}+Q_{T_{\nu,h}},$ (3.6)

as an operator on ${\mathcal{B}}_{i}$, $i=1,2,3$, where $\Pi_{T_{\nu},h}$ is
the one-dimensional projection corresponding to the eigenvalue 1 and
$\Pi_{T_{\nu,h}}Q_{T_{\nu,h}}=Q_{T_{\nu,h}}\Pi_{T_{\nu,h}}=0$.

######  Lemma 3.4.

For each $(\nu,h)\in\Omega$, there exists a unique $H\in{\mathcal{B}}_{1}$
such that

$\Pi_{T_{\nu},h}\phi=H(\nu,h)\int_{{\mathbb{T}}^{d}}\phi,\quad\text{ and
}\quad\|Q^{n}_{T_{\nu,h}}\|_{{\mathcal{B}}_{1}}\leq C\vartheta^{n},$

for some $C>0$. In particular,

$\int H(\nu,h)=1;\quad{\mathcal{L}}_{T_{\nu,h}}H(\nu,h)=H(\nu,h),\textrm{ and
}Q_{T_{\nu,h}}H(\nu,h)=0.$

Moreover,
$H\in{\mathcal{C}}_{*}^{1}(\Omega,{\mathcal{B}}_{1})\cap{\mathcal{C}}^{0}(\Omega,{\mathcal{B}}_{2})\cap
L^{\infty}(\Omega,{\mathcal{B}}_{3})$, 555 Here, ${\mathcal{C}}_{*}^{r}$ means
that the function is only Gâteaux differentiable $r$ times, no assumption on
the continuity of the derivatives is implied. We will use, as usual,
${\mathcal{C}}^{r}$ for Fréchet differentiable functions with continuous
differential. See [11, section 3.2] for the relation between these two
differentials and their properties. and there exists $C_{\\#}>0$, such that,
for each $\varepsilon>0$, $(\nu_{1},h_{1}),(\nu_{2},h_{2})\in\Omega$, for
which $|\nu_{1}-\nu_{2}|+\|h_{1}-h_{2}\|_{{\mathcal{B}}_{0}}\leq\varepsilon$,
we have

$\|H(\nu_{1},h_{1})-H(\nu_{2},h_{2})\|_{{\mathcal{B}}_{2}}\leq
C_{\\#}\varepsilon\ln\varepsilon^{-1}.$

###### Proof.

By (2.2), the Lebesgue measure is a left eigenfunction of
${\mathcal{L}}_{T_{\nu,h}}$ with eigenvalue one. Hence,
$\Pi_{T_{\nu},h}\phi=H(\nu,h)\int_{{\mathbb{T}}^{d}}\phi$ with
${\mathcal{L}}_{T_{\nu,h}}H(\nu,h)=H(\nu,h)$ and $\int H(\nu,h)=1$. Given a
small closed curve $\gamma$ around $1$ functional calculus yields [16]

$\Pi_{T_{\nu},h}=\frac{1}{2\pi
i}\int_{\gamma}(z-{\mathcal{L}}_{T_{\nu,h}})^{-1}dz.$

By Lemma 3.1 and [14, Section 8], or see [13, Theorem 3.3] for more more
precise version, we have

$\Pi_{T_{\nu},h}\in{\mathcal{C}}_{*}^{1}(\Omega,L({\mathcal{B}}_{3},{\mathcal{B}}_{1}))\cap{\mathcal{C}}_{*}^{0}(\Omega,L({\mathcal{B}}_{3},{\mathcal{B}}_{2})).$

Accordingly,

$H\in{\mathcal{C}}_{*}^{1}(\Omega,{\mathcal{B}}_{1})\cap{\mathcal{C}}_{*}^{0}(\Omega,{\mathcal{B}}_{2}).$
(3.7)

Arguing as in the proof of Lemma 2.3, (A5) implies that
$\|H(\nu,h)\|_{{\mathcal{B}}_{3}}\leq C_{\\#}$ for some $C_{\\#}>0$. The bound
on $Q_{T_{\nu,h}}$ follows by Assumption (A6).

To prove the last fact, note that for each $n\in{\mathbb{N}}$, since
${\mathcal{L}}_{T_{\nu,h}}$ has a spectral gap in ${\mathcal{B}}_{2}$, and
using Lemma 3.1,

$\begin{split}&\|H(\nu_{1},h_{1})-H(\nu_{2},h_{2})\|_{{\mathcal{B}}_{2}}=\|{\mathcal{L}}_{T_{\nu_{1},h_{1}}}^{n}H(\nu_{1},h_{1})-{\mathcal{L}}_{T_{\nu_{2},h_{2}}}^{n}H(\nu_{2},h_{2})\|_{{\mathcal{B}}_{2}}\\\
&\leq\|({\mathcal{L}}_{T_{\nu_{1},h_{1}}}^{n}-{\mathcal{L}}_{T_{\nu_{2},h_{2}}}^{n})H(\nu_{1},h_{1})\|_{{\mathcal{B}}_{2}}+\|{\mathcal{L}}_{T_{\nu_{2},h_{2}}}^{n}(H(\nu_{1},h_{1})-H(\nu_{2},h_{2}))\|_{{\mathcal{B}}_{2}}\\\
&\leq\sum_{k=1}^{n-1}\|{\mathcal{L}}_{T_{\nu_{1},h_{1}}}^{k}({\mathcal{L}}_{T_{\nu_{1},h_{1}}}-{\mathcal{L}}_{T_{\nu_{2},h_{2}}}){\mathcal{L}}_{T_{\nu_{2},h_{2}}}^{n-k}H(\nu_{1},h_{1})\|_{{\mathcal{B}}_{2}}+C_{\\#}\nu^{n}\\\
&\leq
C_{\\#}\sum_{k=1}^{n-1}\varepsilon\|{\mathcal{L}}_{T_{\nu_{2},h_{2}}}^{n-k}H(\nu_{1},h_{1})\|_{{\mathcal{B}}_{3}}+C_{\\#}\nu^{n}\leq
C_{\\#}\varepsilon\ln\varepsilon^{-1}\end{split}$

provided we choose $n=c_{\\#}\ln\varepsilon^{-1}$ and $C_{\\#}$ is large
enough. ∎

Next, we provide an estimate needed in the following. Assumptions (A5) and
(A6) allow to write

$\begin{split}\|({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}|_{{\mathbb{V}}}\|_{{\mathcal{B}}_{1}}&\leq\sum_{k=0}^{\infty}\|{\mathcal{L}}_{T_{\nu,h}}^{k}|_{{\mathbb{V}}}\|_{{\mathcal{B}}_{1}}\\\
&\leq\sum_{k=0}^{\bar{n}-1}(C_{*}+K)+\sum_{k=\bar{n}}^{\infty}\vartheta^{n}\leq\bar{n}(C_{*}+K)+(1-\vartheta)^{-1}.\end{split}$

Next, note that Lemma 3.1 and [14, Section 8] implies that there exists
$C_{0}\geq\bar{n}(C_{*}+K)+(1-\vartheta)^{-1}$ such that, for
$i\in\\{0,1,2\\}$.

$\|[({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}-({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu_{0},h_{0}}})^{-1}]\psi\|_{{\mathcal{B}}_{i}}\leq
C_{0}(|\nu-\nu_{0}|+\|h-h_{0}\|_{{\mathcal{B}}_{0}})\|\psi\|_{{\mathcal{B}}_{i+1}}.$

######  Remark 3.5.

The above estiamte and the the definition (2.9), Lemmata 3.3, 3.4 and
Assumption (A2) imply that
$\Theta\in{\mathcal{C}}^{0}(\Omega,{\mathcal{B}}_{1})\bigcup{\mathcal{C}}_{*}^{1}(\Omega,{\mathcal{B}}_{0})$.

######  Lemma 3.6.

$F\in{\mathcal{C}}^{0}(\Omega,{\mathcal{B}}_{2})\cap{\mathcal{C}}^{1}(\Omega,{\mathcal{B}}_{1})\cap{\mathcal{C}}^{2}(\Omega,{\mathcal{B}}_{0})$.
Also, for each $\phi\in{\mathbb{V}}$, the following holds:

$\begin{split}&{D_{h}F}|_{\nu,h}(\phi)=h_{1}-\nu\Theta(\nu,h)\int\alpha\phi\end{split}$

while, if $h\in{\mathcal{B}}_{2}$ and $\phi,\bar{\phi}\in{\mathbb{V}}$, then

$\begin{split}&{D_{h}^{2}F}(\phi,\bar{\phi})=-\Big{\\{}2\sum_{i,j}\left({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}}\right)^{-1}\partial_{x_{i}}\left[{\mathcal{L}}_{T_{\nu,h}}\beta_{i}({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\partial_{x_{j}}\left({\mathcal{L}}_{T_{\nu,h}}\beta_{j}H(\nu,h)\right)\right]\\\
&+({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\sum_{i,j}\partial_{x_{i}}\partial_{x_{j}}\left[{\mathcal{L}}_{T_{\nu,h}}\beta_{i}\beta_{j}H(\nu,h)\right]\Big{\\}}\nu^{2}\int\alpha\phi\int\alpha\bar{\phi}.\end{split}$

Finally, if $h\in{\mathcal{B}}_{1}$,

${\partial_{\nu}F}|_{\nu,h}=-\Theta(\nu,h)\int\alpha h.$

###### Proof.

Gâteaux differentiating ${\mathcal{L}}_{T_{\nu,h}}H(\nu,h)=H(\nu,h)$,
recalling (3.7), (2.9) and using the chain rule [11, Proposition 3.2.12],
yields

$\begin{split}\partial_{h}H(\nu,h)(\phi)&=({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}[(\partial_{h}{\mathcal{L}}_{T_{\nu,h}})H(\nu,h)](\phi)\\\
&=\nu\Theta(\nu,h)\int\alpha\phi,\end{split}$ (3.8)

which yields the formula for the directional derivatives
$\partial_{h}H,\partial_{h}F\in L({\mathcal{B}}_{0},{\mathcal{B}}_{1})$,
which, since they are continuous for $h\in{\mathbb{W}}$ with respect to the
${\mathcal{B}}_{0}$ topology, are the Fréchet derivative ($D_{h}H$ and
$D_{h}F$, repsecrtively ) as well. Using a similar argument to the above, we
also obtain

$\partial_{\nu}H(\nu,h)=\Theta(\nu,h)\int\alpha h.$ (3.9)

By (3.8) and (3.9) and Remark 3.5 it follows that $DH$, as an element of
$L({\mathcal{B}}_{0},{\mathcal{B}}_{0})$, is Gâteaux differentiable, thus
(recalling Lemma 3.4) we obtain
$H\in{\mathcal{C}}^{0}(\Omega,{\mathcal{B}}_{2})\cap{\mathcal{C}}^{1}(\Omega,{\mathcal{B}}_{1})\cap{\mathcal{C}}^{2}_{*}(\Omega,{\mathcal{B}}_{0})$.
To conclude, we must compute $D_{h}^{2}F$. Using (3.4), and (3.9), for every
$\phi,\bar{\phi}\in{\mathcal{C}}^{\infty}$ we have

$\begin{split}&{D_{h}^{2}F}(\phi,\bar{\phi})=-\nu\lim_{\varepsilon\to
0}\frac{\Theta(\nu,h+\varepsilon\bar{\phi})-\Theta(\nu,h)}{\varepsilon}\int\alpha\phi\\\
&=\nu\Big{(}({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}D_{h}{\mathcal{L}}_{T_{\nu,h}}(\bar{\phi})\left[({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\operatorname{div}\left({\mathcal{L}}_{T_{\nu,h}}\beta
H(\nu,h)\right)\right]\\\
&\phantom{=}+({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\operatorname{div}\left[D_{h}{\mathcal{L}}_{T_{\nu,h}}(\bar{\phi})\beta
H(\nu,h)\right]\\\
&\phantom{=}+({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\operatorname{div}\left[{\mathcal{L}}_{T_{\nu,h}}\beta
D_{h}H(\nu,h)(\bar{\phi})\right]\Big{)}\int\alpha\phi.\end{split}$

Then, using (2.9), (3.4) and (3.8), we have

$\begin{split}&{D_{h}^{2}F}(\phi,\bar{\phi})=-\Big{\\{}\sum_{i,j}\left({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}}\right)^{-1}\partial_{x_{i}}{\mathcal{L}}_{T_{\nu,h}}\beta_{i}({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\partial_{x_{j}}{\mathcal{L}}_{T_{\nu,h}}\beta_{j}H(\nu,h)\\\
&+({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\sum_{i,j}\partial_{x_{i}}\partial_{x_{j}}\left[{\mathcal{L}}_{T_{\nu,h}}\beta_{i}\beta_{j}H(\nu,h)\right]\\\
&+\sum_{ij}({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\partial_{x_{i}}{\mathcal{L}}_{T_{\nu,h}}\beta_{i}({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\partial_{x_{j}}{\mathcal{L}}_{T_{\nu,h}}\beta_{j}H(\nu,h)\Big{]}\Big{\\}}\nu^{2}\int\alpha\phi\int\alpha\bar{\phi}.\end{split}$

To conclude, note that, by the above formula and Assumption (A2), the
continuity of $D_{h}^{2}F$ in the ${\mathcal{B}}_{0}$ norm follows from the
continuity of $H$ in the ${\mathcal{B}}_{2}$ norm, which is stated in Lemma
3.4. Hence, the claimed Fréchet differentiability. ∎

##  4\. Charcterization of invariant measures

The basic idea is to apply a quantitative version of the implicit functions
theorem to $F$. See Appendix A for a statement and a proof adapted to our
setting.

######  Lemma 4.1.

For each $(\nu,h)\in\Omega$, $D_{h}F$ is invertible iff
$\nu\int\alpha\Theta(\nu,h)\neq 1$. Moreover, there exist $\bar{C}>1$ such
that, setting $\delta=|\nu-\nu_{0}|+\|h-h_{0}\|_{{\mathcal{B}}_{0}}$, we have

$\begin{split}&\left\|{\mathbbm{1}}-D_{h}F(\nu_{0},h_{0})^{-1}D_{h}F(\nu,h)\right\|_{{\mathcal{B}}_{1}}\leq\bar{C}\delta\ln\delta^{-1}\left\\{\left|1-\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})\right|^{-1}+1\right\\}\\\
&\left\|D_{h}F(\nu_{0},h_{0})^{-1}D_{\nu}F(\nu,h)\right\|_{{\mathcal{B}}_{1}}\leq\bar{C}\left\\{\left|1-\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})\right|^{-1}+1\right\\}.\end{split}$

###### Proof.

If $\nu\int\alpha\Theta(\nu,h)=1$, then, using the explicit formula in Lemma
3.6, we have $D_{h}F\Theta(\nu,h)=0$, hence one implication is trivial. Let us
investigate the other. Note that the tangent space of the domain of $F$ is
${\mathbb{V}}\times{\mathbb{R}}$. Let $h\in{\mathbb{W}}$, and
$\psi\in{\mathbb{V}}$, we want to find $\phi\in{\mathbb{V}}$ such that

$D_{h}F(\phi)=\phi-\nu\Theta(\nu,h)\int\alpha\phi=\psi.$ (4.1)

The above implies that $\phi-\psi=b\Theta(\nu,h)$, for some
$b\in{\mathbb{R}}$, which in turn implies

$b\Theta(\nu,h)=\nu\Theta(\nu,h)\int\alpha(b\Theta(\nu,h)+\psi)$

that is

$b\left(1-\nu\int\alpha\Theta(\nu,h)\right)=\nu\int\alpha\psi$

and

$b=\nu\left(1-\nu\int\alpha\Theta(\nu,h)\right)^{-1}\int\alpha\psi,$

which yields the following unique solution of (4.1)

$\phi=\frac{\nu\Theta(\nu,h)}{1-\nu\int\alpha\Theta(\nu,h)}\int\alpha\psi+\psi.$

Note that, by assumptions (A5) and (A6) there exists $C>0$ such that, for all
$\psi\in{\mathbb{V}}$, $\|\phi\|_{{\mathcal{B}}_{1}}\leq
C\|\psi\|_{{\mathcal{B}}_{1}}$. Accordingly, $D_{h}F$ is invertible and

$(D_{h}F)^{-1}\psi=\frac{\nu\Theta(\nu,h)}{1-\nu\int\alpha\Theta(\nu,h)}\int\alpha\psi+\psi.$
(4.2)

Thus, by Lemma 3.6 and equation (4.2) we can write

$\begin{split}&D_{h}F(\nu_{0},h_{0})^{-1}D_{h}F(\nu,h)\phi=\frac{\nu_{0}\Theta(\nu_{0},h_{0})}{1-\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})}\\\
&\times\int\alpha\left(\phi-\nu\Theta(\nu,h)\int\alpha\phi\right)+\phi-\nu\Theta(\nu,h)\int\alpha\phi\\\
&=\frac{1-\nu\int\alpha\Theta(\nu,h)}{1-\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})}\nu_{0}\Theta(\nu_{0},h_{0})\int\alpha\phi+\phi-\nu\Theta(\nu,h)\int\alpha\phi\\\
&=\frac{\left[\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})-\nu\int\alpha\Theta(\nu,h)\right]}{1-\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})}\nu_{0}\Theta(\nu_{0},h_{0})\int\alpha\phi\\\
&\phantom{=}+(\nu_{0}\Theta(\nu_{0},h_{0})-\nu\Theta(\nu,h))\int\alpha\phi+\phi.\end{split}$

On the other hand, recalling (3.8) and (3.9),

$\begin{split}&\left\|\Theta(\nu_{0},h_{0})-\Theta(\nu,h)\right\|_{{\mathcal{B}}_{1}}\leq\\\
&\leq\Big{\|}\left[({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu_{0},h_{0}}})^{-1}-({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\right]\operatorname{div}(\beta
H(\nu_{0},h_{0}))\|_{{\mathcal{B}}_{1}}\\\
&+\|({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\operatorname{div}\left[\beta
H(\nu_{0},h_{0})-\beta H(\nu,h)\right]\Big{\|}_{{\mathcal{B}}_{1}}\\\ &\leq
C_{0}\|\beta\|_{{\mathcal{C}}^{\bar{r}}}\Big{[}\|H(\nu_{0},h_{0})\|_{{\mathcal{B}}_{3}}\left(|\nu-\nu_{0}|+\|h-h_{0}\|_{{\mathcal{B}}_{0}}\right)\\\
&\phantom{\leq\;}+\|H(\nu_{0},h_{0})-H(\nu,h)\|_{{\mathcal{B}}_{2}}\Big{]}.\end{split}$
(4.3)

Hence, the first inequality of the Lemma follows by Lemma 3.4.

The second inequality follows by equations (4.2) and Lemma 3.6,

$\begin{split}D_{h}F(\nu_{0},h_{0})^{-1}D_{\nu}F(\nu,h)=&\left[\frac{\nu_{0}\Theta(\nu_{0},h_{0})\int\alpha\Theta(\nu,h)}{1-\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})}+\Theta(\nu,h)\right]\\\
&\times\int\alpha h\end{split}$ (4.4)

from which the lemma follows. ∎

###  4.1. Local existence of invariant measures

Our first result is the following.

######  Lemma 4.2.

If $\nu_{0}\in[0,\nu_{*})$, $h_{0}\in{\mathbb{W}}$, $F(\nu_{0},h_{0})=0$ and
$\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})\neq~{}1$, then, setting

$\begin{split}&\delta=\frac{1}{4\bar{C}^{2}\left\\{\left|1-\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})\right|^{-1}+1\right\\}^{2}}\\\
&\delta_{1}=\frac{1}{8\bar{C}^{3}\left\\{\left|1-\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})\right|^{-1}+1\right\\}^{3}},\end{split}$

there exists
${\mathbbm{h}}\in{\mathcal{C}}^{1}((\nu_{0}-\delta_{1},\nu_{0}+\delta_{1})\cap[0,\nu_{*}),{\mathbb{W}})$,
${\mathbbm{h}}(\nu_{0})=h_{0}$, such that all the solutions of $F(\nu,h)=0$ in
the set

$\\{(\nu,h)\in{\mathbb{R}}\times{\mathbb{W}}\;:\;|\nu-\nu_{0}|\leq\delta_{1},\|h-h_{0}\|_{{\mathcal{B}}_{1}}\leq\delta\\}$

belong to the set $\\{{\mathbbm{h}}(s)\\}_{s\in(-\delta_{1},\delta_{1})}$.
Moreover,

${\mathbbm{h}}^{\prime}(s)=-\left[\frac{s\Theta(s,{\mathbbm{h}}(s))\int\alpha\Theta(s,{\mathbbm{h}}(s))}{1-s\int\alpha\Theta(s,{\mathbbm{h}}(s))}+\Theta(s,{\mathbbm{h}}(s))\right]\int\alpha{\mathbbm{h}}(s).$

###### Proof.

Let $\Omega_{1}$ be the space $[0,\nu_{*}]\times{\mathbb{W}}$, where
${\mathbb{W}}$ is endowed with the ${\mathcal{B}}_{1}$ topology. Lemma 3.6
implies a fortiori that $F\in{\mathcal{C}}^{1}(\Omega_{1},{\mathcal{B}}_{1})$.
By Lemma 4.1, if $\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})\neq 1$, we can apply
Theorem A.1 whose hypotheses are satisfied with the claimed
$\delta,\delta_{1}$.666We use the inequality $x\ln x^{-1}\leq\sqrt{x}$.

To conclude, note that, by Theorem A.1 and equation (4.4),

${\mathbbm{h}}^{\prime}(\nu)=-(D_{h}F)^{-1}D_{\nu}F=-\left[\frac{\nu\Theta(\nu,{\mathbbm{h}})\int\alpha\Theta(\nu,{\mathbbm{h}})}{1-\nu\int\alpha\Theta(\nu,{\mathbbm{h}})}+\Theta(\nu,{\mathbbm{h}})\right]\int\alpha{\mathbbm{h}}.$

∎

It remains to analyze what happens if
$\nu_{0}\int\alpha\Theta(h_{0},\nu_{0})=1$. To this end, it is convenient to
define $Z\in{\mathcal{B}}_{1}^{\prime}$ by

$Z(\phi)=\int\alpha\phi.$ (4.5)

Note that, for all $\phi\in{\mathbb{V}}$, using Lemma 3.6, we have

$Z(D_{h_{0}}F(\phi))=\int\alpha\left[\phi-\nu_{0}\Theta(\nu_{0},h_{0})\int\alpha\phi\right]=0.$
(4.6)

Next, we define the change of variables
$\Lambda:{\mathbb{R}}\times{\mathbb{W}}\to{\mathbb{R}}\times{\mathbb{W}}$
defined as

$\Lambda(\tau,\zeta)=(Z(\zeta),\zeta-\tau\nu_{0}\Theta(\nu_{0},h_{0})).$ (4.7)

One can check directly that $\Lambda$ is invertible with inverse given by

$\Lambda^{-1}(\nu,h)=(\nu-Z(h),h-(Z(h)-\nu)\nu_{0}\Theta(\nu_{0},h_{0})).$

Indeed,

$\begin{split}&\Lambda\circ\Lambda^{-1}(\nu,h)\\\
&=(Z(h)-(Z(h)-\nu)\nu_{0}Z(\Theta(\nu_{0},h_{0})),\\\
&\phantom{=\;\;}h-(Z(h)-\nu)\nu_{0}\Theta(\nu_{0},h_{0})-(\nu-Z(h))\nu_{0}\Theta(\nu_{0},h_{0}))=(\nu,h).\end{split}$

Let $(\tau,\zeta)=\Lambda^{-1}(\nu,h)$. Clearly, the equation $F(\nu,h)=0$ is
equivalent to

$\widetilde{F}(\tau,\zeta):=F\circ\Lambda(\tau,\zeta)=0.$ (4.8)

Let $(\tau_{0},\zeta_{0})=\Lambda^{-1}(\nu_{0},h_{0})$. The point of this
change of variable rests in the following result.

######  Lemma 4.3.

If $\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})=1$, then, under the hypothses of
Theorem 2.6, $D_{\zeta}\widetilde{F}(\tau_{0},\zeta_{0})$ is invertible.
Moreover, there exist $\bar{C}>1$ such that, setting
$\delta=|\nu-\nu_{0}|+\|h-h_{0}\|_{{\mathcal{B}}_{0}}$,

$\begin{split}&\left\|{\mathbbm{1}}-D_{\zeta}\widetilde{F}(0,\zeta_{0})^{-1}D_{\zeta}\widetilde{F}(\tau,\zeta)\right\|_{{\mathcal{B}}_{1}}\leq\bar{C}\left[1+\frac{1}{\left|\int\alpha
h_{0}\right|}\right]\delta\ln\delta^{-1}\\\
&\left\|D_{\zeta}\widetilde{F}(0,\zeta_{0})^{-1}D_{\tau}\widetilde{F}(\tau,\zeta)\right\|_{{\mathcal{B}}_{1}}\leq\bar{C}\left[1+\frac{1}{\left|\int\alpha
h_{0}\right|}\right]\delta\ln\delta^{-1}.\end{split}$

###### Proof.

First of all note

$D_{h}F(\Theta(\nu_{0},h_{0}))=0$ (4.9)

so $D_{h}F(\nu_{0},h_{0})$ has a kernel. In addition, equation (4.6) shows
that the range of $D_{h}F(\nu_{0},h_{0})$ has, at least, codimension one.

Next, we compute the derivative of $\widetilde{F}$. Recalling Lemma 3.6,

$\begin{split}D_{\zeta}\widetilde{F}&=(D_{h}F)\circ\Lambda+(D_{\nu}F)\circ\Lambda\otimes
Z\\\ &={\mathbbm{1}}-\left[Z(h)+\nu\right]\Theta(\nu,h)\otimes Z.\end{split}$
(4.10)

Let us show that $D_{\zeta}\widetilde{F}$ is invertible at $(\nu_{0},h_{0})$.
For each $\psi\in{\mathbb{V}}$ we must consider the equation

$\psi=D_{\zeta}\widetilde{F}(\phi)=\phi-\left[Z(h_{0})+\nu_{0}\right]\Theta(\nu_{0},h_{0})Z(\phi).$

Applying $Z$ to the above equation and arguing as in (4.6), we have

$Z(\phi)=\left\\{1-\left[Z(h_{0})+\nu_{0}\right]Z(\Theta(\nu_{0},h_{0}))\right\\}^{-1}Z(\psi)=-\nu_{0}\left\\{Z(h_{0})\right\\}^{-1}Z(\psi).$

Note that $Z(h_{0})\neq 0$ by hypothesis (H1). Thus,

$\phi=\psi-\frac{\nu_{0}\left[Z(h_{0})+\nu_{0}\right]}{Z(h_{0})}\Theta(\nu_{0},h_{0})Z(\psi)=\left[D_{\zeta}\widetilde{F}(\nu_{0},h_{0})\right]^{-1}\hskip-6.0pt(\psi).$
(4.11)

Accordingly, by the bounded inverse Theorem, $D_{\zeta}\widetilde{F}$ is
invertible.

To prove the last inequality of the statement of the Lemma, we recall Lemma
3.6 and we write

$\begin{split}&D_{\zeta}\widetilde{F}(0,\zeta_{0})^{-1}D_{\zeta}\widetilde{F}(\tau,\zeta)(\phi)=\phi-\left[Z(h)+\nu\right]\Theta(\nu,h)Z(\phi)\\\
&\phantom{=}-\frac{\nu_{0}\left[Z(h_{0})+\nu_{0}\right]}{Z(h_{0})}\Theta(\nu_{0},h_{0})\left\\{1-\left[Z(h)+\nu\right]Z(\Theta(\nu,h))\right\\}Z(\phi)\\\
&=\phi-\left[Z(h)-Z(h_{0})+\nu-\nu_{0}\right]\Theta(\nu,h)Z(\phi)\\\
&\phantom{=}-\left[Z(h_{0})+\nu_{0}\right]\left\\{\Theta(\nu,h)-\Theta(\nu_{0},h_{0})\right\\}Z(\phi)\\\
&\phantom{=}+\frac{\nu_{0}\left[Z(h_{0})+\nu_{0}\right]}{Z(h_{0})}\Theta(\nu_{0},h_{0})\left\\{Z(h_{0})Z(\Theta(\nu_{0},h_{0}))-Z(h)Z(\Theta(\nu,h))\right\\}Z(\phi)\\\
&\phantom{=}-\frac{\nu_{0}\left[Z(h_{0})+\nu_{0}\right]}{Z(h_{0})}\Theta(\nu_{0},h_{0})\left\\{\nu_{0}Z(\Theta(\nu_{0},h_{0}))-\nu
Z(\Theta(\nu,h))\right\\}Z(\phi).\end{split}$

The first inequality of the Lemma follows by (4.3) and Lemma 3.4. Let us prove
the second. Note that

$\begin{split}D_{\tau}\widetilde{F}(\tau,\zeta)&=-\nu_{0}D_{h}F(\nu,h)\Theta(\nu_{0},h_{0})\\\
&=-\nu_{0}\Theta(\nu_{0},h_{0})+\nu_{0}\nu\Theta(\nu,h)Z(\Theta(\nu_{0},h_{0}))\\\
&=\nu\Theta(\nu,h)-\nu_{0}\Theta(\nu_{0},h_{0}).\end{split}$

Hence, recalling eqautions (4.11), (4.3) and Lemma 3.4,

$\begin{split}D_{\zeta}\widetilde{F}(0,\zeta_{0})^{-1}&D_{\tau}\widetilde{F}(\tau,\zeta)=D_{\tau}\widetilde{F}(\tau,\zeta)\\\
&-\frac{\nu_{0}\left[Z(h_{0})+\nu_{0}\right]}{Z(h_{0})}\Theta(\nu_{0},h_{0})Z(D_{\tau}\widetilde{F}(\tau,\zeta))\\\
=&\nu\Theta(\nu,h)-\nu_{0}\Theta(\nu_{0},h_{0})\\\
&-\frac{\nu_{0}\left[Z(h_{0})+\nu_{0}\right]}{Z(h_{0})}\Theta(\nu_{0},h_{0})Z(\nu\Theta(\nu,h)-\nu_{0}\Theta(\nu_{0},h_{0})).\end{split}$
(4.12)

The Lemma follows by using (4.3) and Lemma 3.4. ∎

######  Lemma 4.4.

If $\nu_{0}\in(0,\nu_{*})$, $h_{0}\in{\mathbb{W}}$, $F(\nu_{0},h_{0})=0$,
$\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})=~{}{\\!\\!1}$, assumption (H1) holds,
and $\int\alpha h_{0}\not=0$ then there exist $\delta,\delta_{1}>0$, with
$\delta_{1}\leq\nu_{0}^{-1}\|\Theta(\nu_{0},h_{0})\|_{{\mathcal{B}}_{1}}^{-1}\delta$,
and a differentiable map
$(G,{\mathbbm{h}})=:(-\delta_{1},\delta_{1})\to[0,\nu_{*}]\times{\mathbb{W}}$,
$(G(0),{\mathbbm{h}}(0))=(\nu_{0},h_{0})$, such that all the solutions of
$F(\nu,h)=0$ in the set

$\\{(\nu,h)\in{\mathbb{R}}\times{\mathbb{W}}\;:\;|\nu-\nu_{0}|\leq\delta_{1},\|h-h_{0}\|_{{\mathcal{B}}_{1}}\leq\delta\\}$

belong to the set
$\\{G(\tau),{\mathbbm{h}}(\tau)\\}_{\tau\in(-\delta_{1},\delta_{1})}$.
If
$Z(D_{h}^{2}|_{(\nu_{0},h_{0})}F(\Theta(\nu_{0},h_{0}),\Theta(\nu_{0},h_{0}))\neq
0$, then, setting $\nu(\tau)=G(\tau)$,

$\begin{split}&\nu^{\prime}(\tau_{0})=0\\\
&\nu^{\prime\prime}(\tau_{0})=\frac{\nu_{0}^{3}Z\left(D_{h}^{2}F(\Theta(\nu_{0},h_{0}),\Theta(\nu_{0},h_{0}))\right)}{\int\alpha
h_{0}}\neq 0.\end{split}$

###### Proof.

We use the coordinates (4.7). Note that, by the definition (4.7),
$\Lambda\in{\mathcal{C}}^{1}(\Omega_{1},\Omega_{1})$, so
$\widetilde{F}\in{\mathcal{C}}^{1}(\Omega_{1},{\mathcal{B}}_{1})$.777 See the
beginning of the proof of Lemma 4.2 for a definition of $\Omega_{1}$. Note
that there exists $C_{0}>0$ such that, if
$\|\zeta-\zeta_{0}\|_{{\mathcal{B}}_{1}}+\nu_{0}\|\Theta(\nu_{0},h_{0})\|_{{\mathcal{B}}_{1}}|\tau|\leq\delta_{0}$,
then $\|h-h_{0}\|_{{\mathcal{B}}_{1}}\leq C_{0}\delta_{0}$ and
$|\nu-\nu_{0}|\leq C_{0}\delta_{0}$. By Lemma 4.3 we can then choose
$\delta_{0}$ so that

$\left\|{\mathbbm{1}}-D_{\zeta}\widetilde{F}(0,\zeta_{0})^{-1}D_{\zeta}\widetilde{F}(\tau,\zeta)\right\|_{{\mathcal{B}}_{1}}\leq\frac{1}{2},$

and

$\left\|D_{\zeta}\widetilde{F}(0,\zeta_{0})^{-1}D_{\tau}\widetilde{F}(\tau,\zeta)\right\|_{{\mathcal{B}}_{1}}\leq\frac{1}{2}.$

Hence, the Implicit Function Theorem A.1 yields a differentiable function
$\zeta(\tau)$ such that $\widetilde{F}(\tau,\zeta(\tau))=0$ for all
$|\tau|\leq\frac{\delta_{0}}{\nu_{0}\|\Theta(\nu_{0},h_{0})\|_{{\mathcal{B}}_{1}}}$.
The first assertion of the Lemma follows with
$(G(\tau),{\mathbbm{h}}(\tau))=\Lambda(\tau,\zeta(\tau))$. Hence,

$D_{h}F(h^{\prime})+D_{\nu}F\nu^{\prime}=D_{h}F(\zeta^{\prime}-\nu_{0}\Theta(\nu_{0},h_{0}))+D_{\nu}F\nu^{\prime}=0$
(4.13)

which, applying $Z$ and recalling (4.6) and since assumption (H1) implies
$Z(D_{\nu}F)(\nu_{0},h_{0})\neq 0$, gives $\nu^{\prime}(\tau_{0})=0$. Note
that $\nu^{\prime}(\tau_{0})=Z(\zeta^{\prime}(\tau_{0}))$, hence

$\begin{split}0=&D_{h}{F}(\zeta^{\prime}(\tau_{0})-\nu_{0}\Theta(\nu_{0},h_{0}))\\\
=&\zeta^{\prime}(\tau_{0})-\nu_{0}\Theta(\nu_{0},h_{0})-\nu_{0}\Theta(\nu_{0},h_{0})Z(\zeta^{\prime}-\nu_{0}\Theta(\nu_{0},h_{0}))\\\
=&\zeta^{\prime}(\tau_{0})-\nu_{0}\Theta(\nu_{0},h_{0})+\nu_{0}^{2}\Theta(\nu_{0},h_{0})Z(\Theta(\nu_{0},h_{0}))=\zeta^{\prime}(\tau_{0}).\end{split}$

Accordingly,

${\mathbbm{h}}^{\prime}(\tau_{0})=\zeta^{\prime}(\tau_{0})-\nu_{0}\Theta(\nu_{0},h_{0})=-\nu_{0}\Theta(\nu_{0},h_{0}).$
(4.14)

Next, we can differentiate (4.13) considering
$F\in{\mathcal{C}}^{1}(\Omega,{\mathcal{B}}_{0})$ and obtain

$D_{h}^{2}F({\mathbbm{h}}^{\prime},{\mathbbm{h}}^{\prime})+2D^{2}_{\nu,h}F{\mathbbm{h}}^{\prime}\nu^{\prime}+D_{h}F{\mathbbm{h}}^{\prime\prime}+D_{\nu}^{2}F(\nu^{\prime})^{2}+D_{\nu}F\nu^{\prime\prime}=0.$

Setting $\tau=\tau_{0}$ and applying $Z$ yields

$\begin{split}0&=\nu_{0}^{2}Z\left(D_{h}^{2}F(\Theta(\nu_{0},h_{0}),\Theta(\nu_{0},h_{0}))\right)+Z(D_{\nu}F(\nu_{0},h_{0}))\nu^{\prime\prime}(\tau_{0})\\\
&=\nu_{0}^{2}Z\left(D_{h}^{2}F(\Theta(\nu_{0},h_{0}),\Theta(\nu_{0},h_{0}))\right)-\nu_{0}^{-1}\int\alpha
h_{0}\nu^{\prime\prime}(\tau_{0}),\end{split}$

which, by hypothesis, implies $\nu^{\prime\prime}(\tau_{0})\neq 0$. ∎

###### Proof of Theorem 2.6.

The existence of the functions $(G,{\mathbbm{h}})$ follows immediately from
Lemmata 4.2 and 4.4. If
$\nu_{0}\int\alpha\Theta(\nu_{0},{\mathbbm{h}}(\nu_{0}))=1$, then by Lemma 4.4
we have $G(0)=\nu_{0}$, $G^{\prime}(0)=0$ and $G^{\prime\prime}(0)\neq 0$,
hence for each $\tau\in(-\delta,\delta)$, with $\delta$ small enough, we have
that there exists $\tau_{1}<0$ such that $\nu=G(\tau)=G(\tau_{1})$ while, by
(4.14),

${\mathbbm{h}}(\tau)-{\mathbbm{h}}(\tau_{1})=\int_{\tau_{1}}^{\tau}{\mathbbm{h}}^{\prime}(s)ds=(\tau_{1}-\tau)\Theta(\nu_{0},h_{0})+{\mathcal{O}}((\tau-\tau_{1})^{2}),$

where we have used Lemma 4.3. Thus $\tilde{\mathcal{L}}_{\nu}h=h$ has at least
the two solutions $\\{{\mathbbm{h}}(\tau),{\mathbbm{h}}(\tau_{1})\\}$. ∎

###  4.2. Global existence of invariant measures

This section is devoted to the proof of Theorem 2.8, which provides a
criterion that implies the existence of a $\nu$ for which there exists at
least three invariant measures.

To start with, we prove that the solutions in Theorem 2.6 can be extended.

######  Proposition 4.5.

Let $\nu_{0}\in(0,\nu_{*})$ and $h_{0}\in{\mathbb{W}}$ such that
$\tilde{\mathcal{L}}_{\nu_{0}}h_{0}=h_{0}$ and
$\nu_{0}\int\alpha\Theta(\nu_{0},h_{0})\neq 1$. Then there exists an interval
$[\nu_{1},\nu_{2}]\subset[0,\nu_{*}]$, with $\nu_{0}\in(\nu_{1},\nu_{2})$ and
${\mathbbm{h}}\in{\mathcal{C}}^{0}([\nu_{1},\nu_{2}],{\mathcal{B}}_{1})\cap{\mathcal{C}}^{1}((\nu_{1},\nu_{2}),{\mathcal{B}}_{1})$
such that
${\mathcal{L}}_{T_{\nu,{\mathbbm{h}}(\nu)}}{\mathbbm{h}}(\nu)={\mathbbm{h}}(\nu)$.
Moreover, either $\nu_{1}=0$ or
$\nu_{1}\int\alpha\Theta(\nu_{1},{\mathbbm{h}}(\nu_{1}))=1$ and either
$\nu_{2}=\nu_{*}$ or
$\nu_{2}\int\alpha\Theta(\nu_{2},{\mathbbm{h}}(\nu_{2}))=1$.

###### Proof.

We discuss the extension to $\nu_{1}$, the other being identical. Theorem 4.2
implies that there exist $(a,b)\ni\nu_{0}$ and
${\mathbbm{h}}\in{\mathcal{C}}^{1}((a,b),{\mathcal{B}}_{1})$ such that
${\mathcal{L}}_{T_{\nu,{\mathbbm{h}}(\nu)}}{\mathbbm{h}}(\nu)={\mathbbm{h}}(\nu)$.
Lemma 3.4 implies that ${\mathbbm{h}}(a,b)$ is relatively compact in
${\mathcal{B}}_{1}$. Let $h_{*}$ be an accumulation point when $\nu\to a$.
Then, recalling Lemma 3.1,

$h_{*}=\lim_{\nu_{j}\to a}{\mathbbm{h}}_{\nu_{j}}=\lim_{\nu_{j}\to
a}{\mathcal{L}}_{T_{\nu_{j},{\mathbbm{h}}(\nu_{j})}}{\mathbbm{h}}_{\nu_{j}}={\mathcal{L}}_{T_{a,h_{*}}}h_{*}.$

We can then set ${\mathbbm{h}}(a)=h_{*}$ and, if
$a\int\alpha\Theta(a,{\mathbbm{h}}(a))=1$ or $a=0$ we are done, otherwise we
can apply again Theorem 4.2 with $\nu=a$. Clearly this process can stop only
at $0$ or at $\nu_{1}>0$ if
$\nu_{1}\int\alpha\Theta(\nu_{1},{\mathbbm{h}}(\nu_{1}))=1$. This provides a
continuous curve, yet the last formula of Lemma 4.2 implies that
${\mathbbm{h}}\in{\mathcal{C}}^{1}((\nu_{1},\nu_{2}),{\mathcal{B}}_{1})$. ∎

We are now ready to conclude the argument.

###### Proof of Theorem 2.8 .

For $\nu=0$ the map is simply $T$ and by Lemma 2.3 it has a unique invariant
measure $h\in{\mathcal{B}}_{1}$. By Lemma 3.6 $D_{h}F={\mathbbm{1}}$, thus we
can apply Theorem 2.6 and we can extend the solution in an interval
$[0,\nu_{0})$. Then Proposition 4.5 implies that the solution can be extended
to all the interval $[0,\nu_{*}]$ or we lose the invertibility of $D_{h}F$. In
the latter case, we can still extend the solution using Theorem 2.6. In this
way we can define a function
$(G,{\mathbbm{h}}):[0,a)\to[0,\nu_{*})\times{\mathbb{W}}$,
$a\in{\mathbb{R}}\cup\infty$, $G(0)=0$,with the wanted properties. Note that
$G((0,a))\cap\\{0\\}=\emptyset$, since we know that $F(0,h)=0$ has the unique
solution ${\mathbbm{h}}(0)$.

Suppose that there exists $\nu_{1}\in[0,\nu_{*}]$ such that $\sharp
G^{-1}(\nu_{1})=\infty$, then there exists $\\{a_{i}\\}_{i\in{\mathbb{N}}}$
such that
$\widetilde{\mathcal{L}}_{\nu_{1}}{\mathbbm{h}}(a_{i})={\mathbbm{h}}(a_{i})$.
By Lemma 3.4, ${\mathcal{H}}:=\\{{\mathbbm{h}}(a_{i})\\}$ belong to a bounded
set
${\mathbb{W}}_{2}:=\\{h\in{\mathcal{B}}_{2}\cap{\mathbb{W}}\;:\;\|h\|_{{\mathcal{B}}_{2}}\leq
K_{\star}\\}$, so they are relative compact in ${\mathcal{B}}_{1}$ by
Assumption (A4). Let $h_{*}\in{\mathcal{B}}_{1}$ be an accumulation point,
then, by continuity $\widetilde{\mathcal{L}}_{\nu_{1}}h_{*}=h_{*}$. It follows
that the ${\mathcal{B}}_{1}$ closure of ${\mathcal{H}}$ consists of fixed
points on $\widetilde{\mathcal{L}}_{\nu_{1}}$. Let
$B_{\delta}(h)=\\{g\in{\mathbb{W}}\;:\;\|h-g\|_{{\mathcal{B}}_{1}}\leq\delta\\}$.
Then, for each $h\in{\mathbb{W}}_{2}$, if $h\not\in\overline{{\mathcal{H}}}$,
there exists $\delta_{h}>0$ such that
$B_{\delta_{h}}(h)\cap\overline{{\mathcal{H}}}=\emptyset$, while if
$h\in\overline{{\mathcal{H}}}$ then Theorem 2.6 implies that there exists
$\delta_{h}$ such that $h$ is the only solution of
$\widetilde{\mathcal{L}}_{\nu_{1}}g=g$ in $B_{\delta_{h}}(h)$. Since
$\\{B_{\delta_{h}}(h)\\}_{h\in{\mathbb{W}}_{2}}$ is a covering of
${\mathbb{W}}_{2}$, which is relatively compact in ${\mathcal{B}}_{1}$, by
Assumption (A4), we can extract a finite subcover, and this immediately
implies that $\sharp{\mathcal{H}}<\infty$.

It follows that the curve we have constructed cannot go back and forth
infinitely many times in the interval $(0,\nu^{*})$, and it cannot go back to
$\nu=0$, hence it must eventually reach $\nu^{*}$.

To conclude, if $\widetilde{\mathcal{L}}_{\bar{\nu}}h=h$ for some
$\bar{\nu}<\nu^{*}$, and $D_{h}F$ is not invertible, then there are two
possibilities: either $h$ belongs to the curve we constructed starting from
zero or not. In the first case, we have that the curve is described by a
function $(G,{\mathbbm{h}})$, $G(0)=0$, and for some $\tau_{0}$, we have
$G(\tau_{0})=\nu$ and ${\mathbbm{h}}(\tau_{0})=h$. We can assume without loss
of generality that $\tau_{0}$ is the first time $G^{\prime}=0$. Since the
second derivative is non zero, it follows that for $\tau>\tau_{0}$ $G$ is
decreasing. Since it cannot go back to zero it must have another point (we
call them turning points) in which $G^{\prime}=0$ and start to increase again.
Since it can have only finitely many turning points, it must eventually go
back to $\tau_{0}$, and that means that there exists $\epsilon>0$ such that
for $\tau\in(\tau_{0}-\epsilon,\tau_{0})$ we have $\sharp G^{-1}(\tau)\geq 3$.
If instead $G^{\prime}(\tau)\neq 0$ for all $\tau$, then the curve starting
from zero has no turning point, and in the interval $[0,\nu_{*}]$ it provides
an invariant measure for each value of $\tau$. It follows that $h$ does not
belong to such a curve. We can then apply Theorem 2.6 and Proposition 4.5 to
create a curve of solution starting from $h$. Such a curve cannot end at zero
hence it must end, in both directions, at $\nu_{*}$. If we call
$(G_{1},{\mathbbm{h}}_{1})$ such a curve, near $\nu_{*}$ it must have at least
two branches. Hence, together with the curve $(G,{\mathbbm{h}})$, provides
again at least three invariant measures. ∎

##  5\. Local stability and physical measures

Here we study the stability of the invariant measures for a fixed $\nu$. Since
if $\widetilde{\mathcal{L}}_{\nu}h=h$, then $h\in{\mathcal{B}}_{3}$ (see Lemma
3.4) we can study $\widetilde{\mathcal{L}}_{\nu}$ as a function from
${\mathcal{K}}:=\\{h\in{\mathcal{B}}_{3}\;:\;h\geq 0,\int
h=1\\}\subset{\mathcal{B}}_{1}$ to ${\mathcal{B}}_{1}$.888 Note that
${\mathcal{B}}_{3}$ with the ${\mathcal{B}}_{1}$ topology is a normed linear
space, but not a Banach space. Then, for each
$\phi\in{\mathcal{B}}_{1}\cap{\mathbb{V}}$,

$\widetilde{\mathcal{L}}_{\nu}(h+\phi)-\widetilde{\mathcal{L}}_{\nu}(h)=({\mathcal{L}}_{T_{\nu,h+\phi}}-{\mathcal{L}}_{T_{\nu,h}})(h)+{\mathcal{L}}_{T_{\nu,h+\phi}}(\phi).$

By Lemma 3.1 this shows that
$\widetilde{\mathcal{L}}_{\nu}\in{\mathcal{C}}^{1}({\mathcal{K}},{\mathcal{B}}_{1})$
with derivative

${\mathcal{D}}_{h}(\phi):={D}\widetilde{\mathcal{L}}_{\nu}|_{h}(\phi)={\mathcal{L}}_{T_{\nu,h}}\phi-\nu\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta
h\int\alpha\phi.$ (5.1)

It follows that ${\mathcal{D}}_{h}\in L({\mathcal{B}}_{1},{\mathcal{B}}_{1})$
and ${\mathcal{D}}_{h}({\mathbb{V}})\subset{\mathbb{V}}$. Moreover, since
${\mathcal{D}}_{h}$ is a compact perturbation of ${\mathcal{L}}_{T_{\nu,h}}$,
$\sigma_{\operatorname{ess}}({\mathcal{D}}_{h})=\sigma_{\operatorname{ess}}({\mathcal{L}}_{T_{\nu,h}})$.
We can then limit ourselves to the study of the discrete (non-essential)
spectrum. To this end, recall the definitions of $\Theta(z)$ and $\Xi(z)$ in
(2.12),
$\sigma({\mathcal{L}}_{T_{\nu,h}})\setminus\\{1\\}=\sigma(Q_{T_{\nu,h}})$ and
notice that, for $z\not\in\sigma(Q_{T_{\nu,h}})$, we have

$\Theta(z)=-\nu(z{\mathbbm{1}}-Q_{T_{\nu,h}})^{-1}\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta
h.$ (5.2)

######  Proposition 5.1.

Assume that $\Xi(1)\neq 1$, and

$\int\alpha h\neq 0.$

Then $z\in\sigma({\mathcal{D}}_{h})\setminus\sigma({\mathcal{L}}_{T_{\nu,h}})$
iff $\Xi(z)=1$. In addition, $1\in\sigma({\mathcal{D}}_{h})$. Moreover, if $z$
is an eigenvalue with $|z|\geq 1$, then it is a simple eigenvalue, and the
corresponding eigenvector belongs to ${\mathcal{B}}_{2}$. Moreover, if
$\Xi(1)>1$, then there exists $z_{1}\in\sigma({\mathcal{D}}_{h})$ such that
$|z_{1}|>1$; while if $\sup_{\theta}|\Xi(e^{i\theta})|<1$, then there exists
$\kappa<1$ such that
$\sigma({\mathcal{D}}_{h})\setminus\\{1\\}\subset\\{z\in{\mathbb{C}}\;:\;|z|<\kappa\\}$.

###### Proof.

We look for the $\phi\in{\mathcal{B}}_{1}\setminus\\{0\\}$ such that
${\mathcal{D}}_{h}(\phi)=z\phi$ where
$z\in{\mathbb{C}}\setminus\sigma_{\operatorname{ess}}({\mathcal{L}}_{T_{\nu,h}})$.
Then, using (5.1),

$-\nu\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta
h\int\alpha\phi=z\phi-{\mathcal{L}}_{T_{\nu,h}}\phi.$ (5.3)

If $\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta h\equiv 0$, then
${\mathcal{D}}_{h}={\mathcal{L}}_{T_{\nu,h}}$ and the Lemma is trivially true.
We can the assume $\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta
h\not\equiv 0$. If $\int\phi\neq 0$, then we can assume, without loss of
generality, that $\int\phi=1$ and integrating (5.3) yields

$\begin{split}&z=1\\\
&({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})\phi=-\nu\operatorname{div}{\mathcal{L}}_{T_{\nu,h}}\beta
h\int\alpha\phi.\end{split}$ (5.4)

If we set $\phi=h+\hat{\phi}$, $\hat{\phi}\in{\mathbb{V}}$, then by (5.4)

$\hat{\phi}=\Theta(1)\int\alpha\phi$

which, multipying by $\alpha$ and integrating, has a solution only if
$\Xi(1)\neq 1$ and yields

$\begin{split}&\int\alpha\hat{\phi}=[1-\Xi(1)]^{-1}\Xi(1)\int\alpha h\\\
&\int\alpha\phi=[1-\Xi(1)]^{-1}\int\alpha h.\end{split}$

The above implies that

$\phi=h+\Theta(1)[1-\Xi(1)]^{-1}\int\alpha h$ (5.5)

is and eigenvector, and hence $1\in\sigma({\mathcal{D}}_{h})$. We are then
left with the case $\phi\in{\mathbb{V}}$. If
$z\not\in\sigma({\mathcal{L}}_{T_{\nu,h}})$, then equation (5.3) can be
rewritten as

$\phi=\Theta(z)\int\alpha\phi$ (5.6)

which has a nontrivial solution only if $\int\alpha\phi\neq 0$, so we can
assume $\int\alpha\phi=1$. Note that $\Theta(z)\in{\mathcal{B}}_{2}$ since
$h\in{\mathcal{B}}_{3}$. Multiplying equation (5.6) by $\alpha$ and
integrating yields

$\Xi(z)=1.$ (5.7)

On the other hand by (5.1), if $\Xi(z)=1$, then $\phi=\Theta(z)$, and

${\mathcal{D}}_{h}\Theta(z)={\mathcal{L}}_{T_{\nu,h}}\Theta(z)+(z{\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})\Theta(z)=z\Theta(z).$
(5.8)

It follows that, for $z\neq 1$,
$z\in\sigma({\mathcal{D}}_{h})\setminus\sigma({\mathcal{L}}_{T_{\nu,h}})$ iff
$\Xi(z)=1$.

Note that, by the definition (2.12), for $z\in{\mathbb{R}}$, we have
$\Xi(z)\in{\mathbb{R}}$, moreover $\lim_{z\to\infty}\Xi(z)=0$. Accordingly, if
$\Xi(1)>1$, there must be a $z_{1}\in(1,\infty)$ such that $\Xi(z_{1})=1$ and
$z_{1}\in\sigma({\mathcal{D}}_{h})$. On the other hand, if
$\sup_{\theta}|\Xi(e^{i\theta})|<1$, then, since $\Xi$ is holomorphic outside
the unit disk and goes to zero at infinity, we have $\sup_{|z|\geq
1}|\Xi(z)|<1$. Accordingly, all the eventual solutions of (5.7) must be
strictly inside the unit disk. ∎

Before proving our main result, we need one more fact.

######  Lemma 5.2.

$\widetilde{\mathcal{L}}_{\nu}:{\mathcal{K}}\to{\mathcal{K}}$ is weakly
continuous.999 That is, here we view ${\mathcal{K}}$ as a subset of the space
of probability measures.

###### Proof.

If $g_{n}\Longrightarrow g$, weakly, then, for all $\phi\in{\mathcal{C}}^{1}$
we have

$\lim_{n\to\infty}\int\widetilde{\mathcal{L}}_{\nu}(g_{n})\phi-\widetilde{\mathcal{L}}_{\nu}(g)\phi=\lim_{n\to\infty}\int\phi\circ
T_{\nu,g_{n}}g_{n}-\phi\circ T_{\nu,g}g=0.$

The lemma follows since, for each $\tilde{g}\in{\mathcal{K}}$,

$\left|\int\widetilde{\mathcal{L}}_{\nu}(\tilde{g})\phi\right|\leq\|\phi\|_{\infty},$

so the result for all continuous functions can be obtained by approximation. ∎

###### Proof of Theorem 2.12.

If $h$ is visible, let $g\in{\mathcal{C}}^{\infty}$ be such that
$\widetilde{\mathcal{L}}_{\nu}^{n}(g)$ converges weakly to $h$. Then, by
Assumption (A5), for all $n\in{\mathbb{N}}$,

$\|\widetilde{\mathcal{L}}_{\nu}^{n}g\|_{{\mathcal{B}}_{3}}\leq
C_{*}\vartheta^{n}\|g\|_{{\mathcal{B}}_{3}}+K\|g\|_{{\mathcal{B}}_{2}}\leq(C_{*}+K)\|g\|_{{\mathcal{C}}^{\bar{r}}}=:C_{g}.$

Since $\\{\varphi\in{\mathcal{B}}_{3}\;:\;\|\varphi\|_{{\mathcal{B}}_{3}}\leq
C_{g}\\}$ is relatively compact in ${\mathcal{B}}_{2}$ by Assumption (A4),
there exists a sequence $n_{j}$ such that
$\widetilde{\mathcal{L}}_{\nu}^{n_{j}}(g)$ converges to some
$h_{*}\in{\mathcal{B}}_{2}$. Thus, for all $\phi\in{\mathcal{C}}^{\bar{r}}$ we
have, recalling Assumption (A3),

$\int
h_{*}\phi=\lim_{j\to\infty}\int\widetilde{\mathcal{L}}_{\nu}^{n_{j}}(g)\phi=\int\phi
h.$

By the embedding properties of the Banach spaces,
$h=h_{*}\in{\mathcal{B}}_{2}$. The formula for the derivative is then just
(5.1), and, recalling (5.5),

$\begin{split}{\mathcal{D}}_{h}(\phi)&=\left\\{h+\Theta(1)[1-\Xi(1)]^{-1}\int\alpha
h\right\\}\int\phi+{\mathcal{R}}(\phi)\\\
&=\Pi_{0}\phi+{\mathcal{R}}(\phi)\end{split}$ (5.9)

where $\Pi_{0}^{2}=\Pi_{0}$, and
$\Pi_{0}{\mathcal{R}}={\mathcal{R}}\Pi_{0}=0$, in particular
${\mathcal{R}}({\mathcal{B}}_{1})\subset{\mathbb{V}}$.

In the first case of the Theorem we have
$\sigma({\mathcal{R}})\subset\\{z\in{\mathbb{C}}\;:\;|z|<\kappa\\}$. Thus,
there exist $\bar{n}\in{\mathbb{N}}$ such that, for all $n\geq\bar{n}$,
$\|{\mathcal{R}}^{n}\|_{{\mathcal{B}}_{1}}\leq\frac{1}{2}$. It follows that,
for $g\in{\mathbb{V}}$, small enough in $\|\cdot\|_{{\mathcal{B}}_{1}}$, we
have101010 By the chain rule for the Frechét derivative, [11, Section 3.2], we
have $D_{h}(\widetilde{\mathcal{L}}_{\nu}^{n})={\mathcal{D}}_{h}^{n}$.

$\begin{split}\|\widetilde{\mathcal{L}}_{\nu}^{\bar{n}}(h+g)-h\|_{{\mathcal{B}}_{1}}&=\|\widetilde{\mathcal{L}}_{\nu}^{\bar{n}}(h+g)-\widetilde{\mathcal{L}}_{\nu}^{\bar{n}}(h)\|_{{\mathcal{B}}_{1}}\\\
&\leq\|{\mathcal{D}}_{h}^{\bar{n}}(g)\|_{{\mathcal{B}}_{1}}+o(\|g\|_{{\mathcal{B}}_{1}})=\|{\mathcal{R}}^{\bar{n}}(g)\|_{{\mathcal{B}}_{1}}+o(\|g\|_{{\mathcal{B}}_{1}})\\\
&\leq\frac{3}{4}\|g\|_{{\mathcal{B}}_{1}}.\end{split}$

Accordingly, there exists $\delta>0$ such that, for all
$\|g\|_{{\mathcal{B}}_{1}}\leq\delta$,

$\lim_{n\to\infty}\|\widetilde{\mathcal{L}}_{\nu}^{n}(h+g)-h\|_{{\mathcal{B}}_{1}}=0.$
(5.10)

Since ${\mathcal{C}}^{\infty}\subset{\mathcal{B}}_{1}$, by (5.10) and (A3), it
follows that for an open set of densities in ${\mathcal{C}}^{\infty}$ the
sequence of measures converges weakly to $h$. Hence, $h$ is a physical
measure.

On the contrary, if
$\sigma({\mathcal{D}}_{h})\setminus\\{z\in{\mathbb{C}}\;:\;|z|\leq
1\\}\neq\emptyset$, by the spectral decomposition, we can further refine the
decomposition in (5.9) as

${\mathcal{R}}(\phi)=A(\phi)+B(\phi),$ (5.11)

$AB=BA=0$, $A$ is finite rank and
$\sigma(A)\subset\\{z\in{\mathbb{C}}\;:\;|z|>\kappa^{-1}\\}\cup\\{0\\}$,
$\kappa<1$, while $\sigma(B)\subset\\{z\in{\mathbb{C}}\;:\;|z|<\kappa\\}$. Let
${\mathbb{V}}_{A}=\operatorname{Range(A)}$,
${\mathbb{V}}_{B}=\operatorname{Range(B)}$, and
${\mathbb{V}}_{0}=\operatorname{Range(\Pi_{0})}$, then
${\mathcal{B}}_{1}={\mathbb{V}}_{A}\oplus{\mathbb{V}}_{0}\oplus{\mathbb{V}}_{B}$.
Next, notice that equation (5.1) implies, for each
$\tilde{g},\tilde{\phi}\in{\mathcal{B}}_{1}$,

$\int
D_{\tilde{g}}\widetilde{\mathcal{L}}_{\nu}(\tilde{\phi})=\int\tilde{\phi},$
(5.12)

which implies that ${\mathbb{V}}={\mathbb{V}}_{A}\oplus{\mathbb{V}}_{B}$.

Let $\Pi_{A}\in L({\mathcal{B}}_{1},{\mathcal{B}}_{1})$ the the projection of
on ${\mathbb{V}}_{A}$ along ${\mathbb{V}}_{0}\oplus{\mathbb{V}}_{B}$, i.e.
$\Pi_{A}^{2}=\Pi_{A}$ and $\Pi_{A}{\mathcal{D}}_{h}=A$. Fix
$\kappa_{1}\in(\kappa,1)$ and let $\bar{n}\in{\mathbb{N}}$ be such that
$\|A^{\bar{n}}g\|_{{\mathcal{B}}_{1}}\geq\kappa_{1}^{-\bar{n}}\|g\|_{{\mathcal{B}}_{1}}$,
for all $g\in{\mathbb{V}}_{A}$, and
$\|B^{\bar{n}}\|_{{\mathcal{B}}_{1}}\leq\kappa_{1}^{\bar{n}}$. Remark that
$\widetilde{\mathcal{L}}_{\nu}{\mathcal{K}}\subset{\mathcal{K}}$, so we can
compose it and $D_{h}\widetilde{\mathcal{L}}_{\nu}^{n}={\mathcal{D}}_{h}^{n}$.
Moreover, by Lemma 2.3, we have
$\widetilde{\mathcal{L}}_{\nu}^{\bar{n}}{\mathbb{W}}\subset{\mathbb{W}}$.

We argue by contradiction. Assume that $h$ is physical. That is, there exists
a ${\mathcal{C}}^{\infty}$ open set ${\mathcal{A}}$ that converges weakly to
$h$. Note that, by eventually restricting ${\mathcal{A}}$, we can assume that
$\sup_{g\in{\mathcal{A}}}\|g\|_{{\mathcal{C}}^{\bar{r}}}<\infty$.

######  Lemma 5.3.

For each $\nu\in[0,\nu_{*}]$ and $g\in{\mathcal{A}}$, we have

$\lim_{n\to\infty}\|\widetilde{\mathcal{L}}_{\nu}^{n}g-h\|_{{\mathcal{B}}_{2}}=0.$

Moreover, there exists $C_{\\#}>0$ such that, for each $\varepsilon>0$, there
exists $m_{\varepsilon}\in{\mathbb{N}}$ such that, for each $n\geq
m_{\varepsilon}$,
$\widetilde{\mathcal{L}}_{\nu}^{n}{\mathcal{A}}\subset{\mathbb{B}}_{\varepsilon}$,
where

${\mathbb{B}}_{\varepsilon}:=\\{g\in{\mathcal{B}}_{2}\cap{\mathbb{W}}\;:\;\|g-h\|_{{\mathcal{B}}_{2}}\leq
2C_{\\#}C_{2}\varepsilon\ln\varepsilon^{-1}\\}.$

###### Proof.

By (2.5)
$C_{1}:=\sup_{g\in{\mathcal{A}}}\|g\|_{{\mathcal{B}}_{3}}\leq\sup_{g\in{\mathcal{A}}}\|g\|_{{\mathcal{C}}^{\bar{r}}}<\infty$.
By Assumption (A5), for all $n\in{\mathbb{N}}$ and $g\in{\mathcal{A}}$,

$\|\widetilde{\mathcal{L}}_{\nu}^{n}g\|_{{\mathcal{B}}_{3}}\leq
C_{*}\vartheta^{n}\|g\|_{{\mathcal{B}}_{3}}+K\|g\|_{{\mathcal{B}}_{2}}\leq(C_{*}+K)C_{1}.$

Let $C_{2}=\max\\{\|h\|_{{\mathcal{B}}_{3}},(C_{*}+K)C_{1}\\}$. For each
$\varepsilon>0$, we defined the weak $\varepsilon$-neighborhood

${\mathcal{U}}_{\varepsilon}=\left\\{g\in{\mathcal{C}}^{0}\;:\;\left|\int\alpha(h-g)\right|<\varepsilon\right\\}.$

Next, by our hypothesis, for each $\varepsilon>0$ there exists
$n_{\varepsilon}\in{\mathbb{N}}$ such that
$\widetilde{\mathcal{L}}_{\nu}^{n}({\mathcal{A}})\in{\mathcal{U}}_{\varepsilon/2}$,
for $n\geq n_{\varepsilon}$. Then, for all $m\geq n_{\varepsilon}$,
$n\in\\{0,\dots,C_{4}\ln\varepsilon^{-1}\\}$ and
$g\in\widetilde{\mathcal{L}}_{\nu}^{m}({\mathcal{A}})$, let
$g_{n}=\widetilde{\mathcal{L}}_{\nu}^{m+n}g$ and recalling equation (5.1),

$\begin{split}&\|g_{n}-h\|_{{\mathcal{B}}_{2}}=\|\widetilde{\mathcal{L}}_{\nu}(g_{n-1})-\widetilde{\mathcal{L}}_{\nu}(h)\|_{{\mathcal{B}}_{1}}\leq\int_{0}^{1}dt\Big{\|}{\mathcal{L}}_{T_{\nu,h+t(g_{n-1}-h)}}(g_{n-1}-h)\\\
&-\nu\operatorname{div}{\mathcal{L}}_{T_{\nu,h+t(g_{n-1}-h)}}\beta\left[h+t(g_{n-1}-h)\right]\int\alpha(g_{n-1}-h)\Big{\|}_{{\mathcal{B}}_{2}}\\\
&\leq\int_{0}^{1}dt\Big{\|}{\mathcal{L}}_{T_{\nu,h+t(g_{n-1}-h)}}(g_{n-1}-h)\Big{\|}_{{\mathcal{B}}_{2}}+C_{\\#}C_{2}\varepsilon\\\
&\leq\int_{[0,1]^{n}}dt_{1}\cdots
dt_{n}\Big{\|}{\mathcal{L}}_{T_{\nu,h+t_{1}(g_{n-1}-h)}}{\mathcal{L}}_{T_{\nu,h+t_{2}(g_{n-2}-h)}}\cdots{\mathcal{L}}_{T_{\nu,h+t_{n}(g-h)}}(g-h)\Big{\|}_{{\mathcal{B}}_{2}}\\\
&\phantom{=}+C_{\\#}C_{2}\varepsilon n.\end{split}$

Then, for $\varepsilon$ small enough, by Assumption (A6) we have

$\|g_{n}-h\|_{{\mathcal{B}}_{2}}\leq
2\vartheta^{n}C_{2}+C_{\\#}C_{2}n\varepsilon.$

It follows that, choosing $C_{4}$ large enough, we have

$\|g_{C_{4}\ln\varepsilon^{-1}}-h\|_{{\mathcal{B}}_{2}}\leq
2C_{\\#}C_{2}\varepsilon\ln\varepsilon^{-1}.$

Hence, for all $k\geq n_{\varepsilon}+C_{4}\ln\varepsilon^{-1}$ we have
$\widetilde{\mathcal{L}}_{\nu}^{k}({\mathcal{A}})\subset{\mathbb{B}}_{\varepsilon}$.
∎

Let
$\widetilde{\mathbb{B}}_{\varepsilon}={\mathbb{B}}_{\varepsilon}\bigcap\widetilde{\mathcal{L}}_{\nu}^{m_{\varepsilon}}{\mathcal{A}}$,
where $m_{\varepsilon}$ and ${\mathbb{B}}_{\varepsilon}$ are defined in the
statement of Lemma 5.3. We will need the following approximation result.

######  Lemma 5.4.

For each $\nu\in[0,\nu_{*}]$ and $C_{\star}>0$ there exists
${\overline{m}}\in{\mathbb{N}}$, such that, for each $m\geq{\overline{m}}$
there exists $\varepsilon_{0}>0$ such that, for each
$\varepsilon\in(0,\varepsilon_{0})$,
$g\in\widetilde{\mathbb{B}}_{\varepsilon}$, and $\phi\in{\mathcal{B}}_{1}$, we
have

$\|D_{g}\widetilde{\mathcal{L}}_{\nu}^{m}\phi-{\mathcal{D}}_{h}^{m}\phi\|_{{\mathcal{B}}_{1}}\leq\frac{1}{4C_{\star}}\|\phi\|_{{\mathcal{B}}_{1}}.$

###### Proof.

To start with, note that, recalling (5.1), there exists $K>0$ such that, for
$i\in\\{0,1\\}$,

$\sup_{g\in{\mathbb{B}}_{\varepsilon}}\|{\mathcal{D}}_{g}\|_{{\mathcal{B}}_{i}}\leq
K.$

If $\tilde{g}\in\widetilde{\mathbb{B}}_{\varepsilon}$, then Assumptions (A2),
(A3) together with equations (3.4) imply

$\begin{split}\left\|{\mathcal{L}}_{T_{\nu,\tilde{g}}}\psi-{\mathcal{L}}_{T_{\nu,h}}\psi\right\|_{{\mathcal{B}}_{i}}&\leq\int_{0}^{1}\left\|\nu\operatorname{div}{\mathcal{L}}_{\nu,t\tilde{g}+(1-t)h}\beta\psi\int\alpha(\tilde{g}-h)\right\|_{{\mathcal{B}}_{i}}\\\
&\leq|\nu|C_{\\#}\|\psi\|_{{\mathcal{B}}_{i+1}}\|\tilde{g}-h\|_{{\mathcal{B}}_{0}}.\end{split}$
(5.13)

Let $g_{m}:=\widetilde{\mathcal{L}}_{\nu}^{m}(g)$. Recalling (A2), (A3) and
(5.1), we have, for all $\phi\in{\mathcal{B}}_{i}$, for $i\in\\{0,1\\}$, and
$L\in L({\mathcal{B}}_{0},{\mathcal{B}}_{0})\cap
L({\mathcal{B}}_{1},{\mathcal{B}}_{1})$, such that
$\|L\|_{{\mathcal{B}}_{i}}\leq B$ for some constant $B>0$,111111 We will see
shortly the convenience of allowing an arbitrary operator $L$.

$\displaystyle\|L\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m}\phi-{\mathcal{D}}_{h}^{m}\phi\right)\|_{{\mathcal{B}}_{i}}\leq\left\|L\left({\mathcal{L}}_{T_{\nu,g_{m-1}}}\\!\\!D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{L}}_{T_{\nu,h}}{\mathcal{D}}_{h}^{m-1}\phi\right)\right\|_{{\mathcal{B}}_{i}}$
$\displaystyle+C_{\\#}|\nu|B\left\|{\mathcal{L}}_{T_{\nu,g_{m-1}}}\beta
g_{m-1}\int\alpha
D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{L}}_{T_{\nu,h}}\beta
h\int\alpha{\mathcal{D}}_{h}^{m-1}\phi\right\|_{{\mathcal{B}}_{i+1}}$
$\displaystyle\leq\left\|L\left({\mathcal{L}}_{T_{\nu,g_{m-1}}}D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{L}}_{T_{\nu,h}}{\mathcal{D}}_{h}^{m-1}\phi\right)\right\|_{{\mathcal{B}}_{i}}$
$\displaystyle+C_{\\#}B|\nu|K^{m-1}\left\|{\mathcal{L}}_{T_{\nu,g_{m-1}}}\beta(g_{m-1}-h)\right\|_{{\mathcal{B}}_{i+1}}\|\phi\|_{{\mathcal{B}}_{0}}$
$\displaystyle+C_{\\#}B|\nu|K^{m-1}\left\|{\mathcal{L}}_{T_{\nu,g_{m-1}}}\beta
h-{\mathcal{L}}_{T_{\nu,h}}\beta
h)\right\|_{{\mathcal{B}}_{i+1}}\|\phi\|_{{\mathcal{B}}_{0}}$ (5.14)
$\displaystyle+C_{\\#}B|\nu|\left\|{\mathcal{L}}_{T_{\nu,h}}\beta
h\right\|_{{\mathcal{B}}_{i+1}}\left|\int\alpha\left[D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{D}}_{h}^{m-1}\phi\right]\right|$
$\displaystyle\leq\left\|L{\mathcal{L}}_{T_{\nu,g_{m-1}}}\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{D}}_{h}^{m-1}\phi\right)\right\|_{{\mathcal{B}}_{i}}$
$\displaystyle+B\left\|\left({\mathcal{L}}_{T_{\nu,g_{m-1}}}-{\mathcal{L}}_{T_{\nu,h}}\right){\mathcal{D}}_{h}^{m-1}\phi\right\|_{{\mathcal{B}}_{i}}$
$\displaystyle+C_{\\#}B|\nu|\left\\{K^{m}\varepsilon\ln\varepsilon^{-1}C_{2}\|\phi\|_{{\mathcal{B}}_{1}}+\|D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{D}}_{h}^{m-1}\phi\|_{{\mathcal{B}}_{0}}\right\\}.$

Hence, setting
$A_{*}=\sup_{\tilde{g}\in\widetilde{\mathbb{B}}_{\varepsilon}}\max\left\\{\left\|{\mathcal{L}}_{T_{\nu,\tilde{g}}}\right\|_{{\mathcal{B}}_{0}},\left\|{\mathcal{L}}_{T_{\nu,\tilde{g}}}\right\|_{{\mathcal{B}}_{1}}\right\\}$,
we can use equation (5.14), with $L={\mathbbm{1}}$, and (5.13) to write

$\begin{split}\|D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{D}}_{h}^{m-1}\phi\|_{{\mathcal{B}}_{0}}\leq&B(A_{*}+C_{\\#}|\nu|)\left\|D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-2}\phi-{\mathcal{D}}_{h}^{m-2}\phi\right\|_{{\mathcal{B}}_{0}}\\\
&+C_{\\#}B|\nu|\left\\{K^{m-1}\varepsilon\ln\varepsilon^{-1}C_{2}\|\phi\|_{{\mathcal{B}}_{1}}\right\\}.\end{split}$

Choosing $K\geq B(A_{*}+C_{\\#}|\nu|)$ large enough and iterating the above
equation yields

$\|D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{D}}_{h}^{m-1}\phi\|_{{\mathcal{B}}_{0}}\leq
C_{3}mK^{2m}\varepsilon\ln\varepsilon^{-1}\|\phi\|_{{\mathcal{B}}_{1}}.$
(5.15)

for some $C_{3}$ large enough. Using (5.15) in (5.14) yields

$\begin{split}\left\|L\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m}\phi-{\mathcal{D}}_{h}^{m}\phi\right)\right\|_{{\mathcal{B}}_{1}}\leq&\left\|L{\mathcal{L}}_{T_{\nu,g_{m-1}}}\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{D}}_{h}^{m-1}\phi\right)\right\|_{{\mathcal{B}}_{1}}\\\
&+B\left\|\left({\mathcal{L}}_{T_{\nu,g_{m-1}}}-{\mathcal{L}}_{T_{\nu,h}}\right){\mathcal{D}}_{h}^{m-1}\phi\right\|_{{\mathcal{B}}_{1}}\\\
&+C_{3}B(m+1)K^{2m}\varepsilon\ln\varepsilon^{-1}C_{2}\|\phi\|_{{\mathcal{B}}_{1}}.\end{split}$
(5.16)

Note that, calling $\Pi_{B}$ the eigenprojection on ${\mathbb{V}}_{B}$ along
${\mathbb{V}}_{0}\oplus{\mathbb{V}}_{A}$, Proposition 5.1 implies that
$({\mathbbm{1}}-\Pi_{B})$ is fine rank and $({\mathbbm{1}}-\Pi_{B})\in
L({\mathcal{B}}_{1},{\mathcal{B}}_{2})$. Moreover, by definition, there exists
$C_{p}>0$ such that

$\|\Pi_{B}{\mathcal{D}}_{h}^{m}\|_{{\mathcal{B}}_{1}}=\|\Pi_{B}{\mathcal{D}}_{h}^{m}\Pi_{B}\|_{{\mathcal{B}}_{1}}=\|\left(\Pi_{B}{\mathcal{D}}_{h}\Pi_{B}\right)^{m}\|_{{\mathcal{B}}_{1}}\leq
C_{p}\kappa_{1}^{m}$

for all $m\in{\mathbb{N}}$. In addition, for $C_{p}$ large enough,
$\|{\mathbbm{1}}-\Pi_{B}\|_{{\mathcal{B}}_{1}\to{\mathcal{B}}_{2}}\leq C_{p}$.
Hence, equation (5.13) implies

$\begin{split}\left\|\left({\mathcal{L}}_{T_{\nu,g_{m-1}}}-{\mathcal{L}}_{T_{\nu,h}}\right){\mathcal{D}}_{h}^{m-1}\phi\right\|_{{\mathcal{B}}_{1}}\leq&\left\|\left({\mathcal{L}}_{T_{\nu,g_{m-1}}}-{\mathcal{L}}_{T_{\nu,h}}\right)({\mathbbm{1}}-\Pi_{B}){\mathcal{D}}_{h}^{m-1}\phi\right\|_{{\mathcal{B}}_{1}}\\\
&+\left\|\left({\mathcal{L}}_{T_{\nu,g_{m-1}}}-{\mathcal{L}}_{T_{\nu,h}}\right)\Pi_{B}{\mathcal{D}}_{h}^{m-1}\phi\right\|_{{\mathcal{B}}_{1}}\\\
\leq&\kappa_{1}^{m}C_{p}\varepsilon\ln\varepsilon^{-1}\|\phi\|_{{\mathcal{B}}_{1}}+2A_{*}\kappa_{1}^{-m}\left\|\phi\right\|_{{\mathcal{B}}_{1}}.\end{split}$

Using the above fact in equation (5.16) yields

$\begin{split}\left\|L\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m}\phi-{\mathcal{D}}_{h}^{m}\phi\right)\right\|_{{\mathcal{B}}_{1}}\leq&\left\|L{\mathcal{L}}_{T_{\nu,g_{m-1}}}\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-1}\phi-{\mathcal{D}}_{h}^{m-1}\phi\right)\right\|_{{\mathcal{B}}_{1}}\\\
&+B\left(\kappa_{1}^{m}C_{p}\varepsilon\ln\varepsilon^{-1}+2A_{*}\kappa_{1}^{-m}\right)\left\|\phi\right\|_{{\mathcal{B}}_{1}}\\\
&+C_{3}B(m+1)K^{2m}\varepsilon\ln\varepsilon^{-1}C_{2}\|\phi\|_{{\mathcal{B}}_{1}}.\end{split}$

To conlude we iterate the above equation $k$ times, starting with
$L={\mathbbm{1}}$, and use Assumption (A5) to bound uniformly
${\mathcal{L}}_{T_{\nu,g_{m-1}}}\cdots{\mathcal{L}}_{T_{\nu,g_{m-j}}}$, for
each $j\in\\{1,\dots,k\\}$. By doing this we obtain, for some $C_{*}>0$,

$\begin{split}\left\|\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m}\phi-{\mathcal{D}}_{h}^{m}\phi\right)\right\|_{{\mathcal{B}}_{1}}\leq&\left\|{\mathcal{L}}_{T_{\nu,g_{m-1}}}\cdots{\mathcal{L}}_{T_{\nu,g_{m-k}}}\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-k}\phi-{\mathcal{D}}_{h}^{m-k}\phi\right)\right\|_{{\mathcal{B}}_{1}}\\\
&+C_{*}\left(\kappa_{1}^{m}\varepsilon\ln\varepsilon^{-1}+\kappa_{1}^{-m+k}\right)\left\|\phi\right\|_{{\mathcal{B}}_{1}}\\\
&+C_{*}(m+1)K^{2m}\varepsilon\ln\varepsilon^{-1}\|\phi\|_{{\mathcal{B}}_{1}}.\end{split}$

Note that (5.12) implies
$D_{g}\widetilde{\mathcal{L}}_{\nu}^{m-k}\phi-{\mathcal{D}}_{h}^{m-k}\phi\in{\mathbb{V}}$.
Accordingly, Assumption (A6) implies, for $k\geq\bar{n}$,

$\begin{split}\left\|\left(D_{g}\widetilde{\mathcal{L}}_{\nu}^{m}\phi-{\mathcal{D}}_{h}^{m}\phi\right)\right\|_{{\mathcal{B}}_{1}}\leq&2\vartheta^{k}K^{m-k}\left\|\phi\right\|_{{\mathcal{B}}_{1}}\\\
&+C_{*}\left(\kappa_{1}^{m}\varepsilon\ln\varepsilon^{-1}+\kappa_{1}^{-m+k}\right)\left\|\phi\right\|_{{\mathcal{B}}_{1}}\\\
&+C_{*}(m+1)K^{2m}\varepsilon\ln\varepsilon^{-1}\|\phi\|_{{\mathcal{B}}_{1}}.\end{split}$

The Lemma follows by choosing first $m-k$ large, then $k$ large and, finally,
$\varepsilon$ small enough. ∎

Recalling the definition of $\Pi_{0}$ and $\Pi_{A}$ just after equation
(5.12), setting $\Pi_{B}={\mathbbm{1}}-\Pi_{A}-\Pi_{0}$. we define,

$C_{\star}:=\max\\{\|{\mathbbm{1}}-\Pi_{A}\|_{{\mathcal{B}}_{1}},\|\Pi_{0}\|_{{\mathcal{B}}_{1}},\|\Pi_{B}\|_{{\mathcal{B}}_{1}},1\\},$

and $\overline{m}\in{\mathbb{N}}$ be so that
$\|\Pi_{B}{\mathcal{D}}_{h}^{\overline{m}}\Pi_{B}\|_{{\mathcal{B}}_{1}}\leq\frac{1}{7C_{\star}}$,
and $\|{\mathcal{D}}_{h}^{\overline{m}}\tilde{g}\|_{{\mathcal{B}}_{1}}\geq
7C_{*}\|\tilde{g}\|_{{\mathcal{B}}_{1}}$, for each
$\tilde{g}\in\operatorname{Range}(\Pi_{A})$. Let
$g\in\widetilde{\mathbb{B}}_{\varepsilon}$. Since
${\mathcal{L}}^{n_{\varepsilon}}{\mathcal{A}}$ is open in the
${\mathcal{C}}^{\infty}$ topology, we have
$(g+{\mathcal{C}})\cap{\mathcal{L}}^{m_{\varepsilon}}{\mathcal{A}}\neq\emptyset$
where

${\mathcal{C}}=\left\\{\xi\in{\mathbb{B}}_{\varepsilon}\;:\;\|({\mathbbm{1}}-\Pi_{A})\xi\|_{{\mathcal{B}}_{1}}\leq\frac{1}{2}\|\Pi_{A}\xi\|_{{\mathcal{B}}_{1}}\right\\},$

$\Pi_{A}$ being the projector on the expanding subspace of
${\mathcal{D}}_{h}$. We can then consider
$\gamma:=g+\xi\in{\mathcal{L}}^{m_{\varepsilon}}{\mathcal{A}}$, with
$\xi\in{\mathcal{C}}$. For all $n\in{\mathbb{N}}$ set
$g_{n}=\widetilde{\mathcal{L}}_{\nu}^{n{{\overline{m}}}}g$,
$\gamma_{n}=\widetilde{\mathcal{L}}_{\nu}^{n{\overline{m}}}\gamma$ and
$\xi_{n}=g_{n}-\gamma_{n}$. By Lemma 5.3
$g_{n},\gamma_{n}\in{\mathbb{B}}_{\varepsilon}$ for all $n\in{\mathbb{N}}$,
hence so is $g_{n}+t\xi_{n}$ for all $t\in[0,1]$. Our aim is to prove, by
induction, that $\xi_{n}\in{\mathcal{C}}$ for all $n\in{\mathbb{N}}$. If
$\xi_{n}\in{\mathcal{C}}$ , then we can apply Lemma 5.4 with $\varepsilon$
small enough to obtain

$\begin{split}&\|({\mathbbm{1}}-\Pi_{A})\xi_{n+1}\|_{{\mathcal{B}}_{1}}=\left\|(\Pi_{0}+\Pi_{B})\int_{0}^{1}D_{g_{n}+\xi_{n}t}\widetilde{\mathcal{L}}_{\nu}^{\,\overline{m}}\xi_{n}dt\right\|_{{\mathcal{B}}_{1}}\\\
&\leq\|{\mathcal{D}}^{\overline{m}}_{h}(\Pi_{0}+\Pi_{B})\xi_{n}\|_{{\mathcal{B}}_{1}}+\frac{1}{4}\|\xi_{n}\|_{{\mathcal{B}}_{1}}\leq\left\\{C_{\star}+\frac{1}{7C_{\star}}\right\\}\|\xi_{n}\|_{{\mathcal{B}}_{1}}+\frac{1}{4}\|\xi_{n}\|_{{\mathcal{B}}_{1}}\\\
&\leq\left(2C_{\star}+\frac{3}{8}\right)\|\Pi_{A}\xi_{n}\|_{{\mathcal{B}}_{1}}\leq
3C_{\star}\|\Pi_{A}\xi_{n}\|_{{\mathcal{B}}_{1}}.\end{split}$ (5.17)

On the other hand,

$\begin{split}\|\Pi_{A}\xi_{n+1}\|_{{\mathcal{B}}_{1}}&=\left\|\Pi_{A}\int_{0}^{1}D_{g_{n}+\xi_{n}t}\widetilde{\mathcal{L}}_{\nu}^{\,\overline{m}}\xi_{n}\right\|_{{\mathcal{B}}_{1}}\\\
&\geq\|{\mathcal{D}}_{h}^{\overline{m}}\Pi_{A}\xi_{n}\|_{{\mathcal{B}}_{1}}-\frac{1}{4}\|\xi_{n}\|_{{\mathcal{B}}_{1}}\geq
7C_{\star}\|\Pi_{A}\xi_{n}\|_{{\mathcal{B}}_{1}}-\frac{1}{4}\|\xi_{n}\|_{{\mathcal{B}}_{1}}\\\
&\geq 6C_{\star}\|\Pi_{A}\xi_{n}\|_{{\mathcal{B}}_{1}}.\end{split}$ (5.18)

It follows

$\|({\mathbbm{1}}-\Pi_{A})\xi_{n+1}\|_{{\mathcal{B}}_{1}}\leq\frac{1}{2}\|\Pi_{A}\xi_{n+1}\|_{{\mathcal{B}}_{1}},$

that is $\xi_{n+1}\in{\mathcal{C}}$. In addition,

$\|\xi_{n}\|_{{\mathcal{B}}_{1}}\geq\|\Pi_{A}\xi_{n}\|_{{\mathcal{B}}_{1}}-\|({\mathbbm{1}}-\Pi_{A})\xi_{n}\|_{{\mathcal{B}}_{1}}\geq\frac{1}{2}\|\Pi_{A}\xi_{n}\|_{{\mathcal{B}}_{1}}\geq\frac{(6C_{*})^{n}}{3}\|\xi_{n}\|_{{\mathcal{B}}_{1}},$

contradicting the fact that
$\widetilde{\mathcal{L}}_{\nu}^{n}\gamma\in{\mathbb{B}}_{\varepsilon}$ for all
$n\in{\mathbb{N}}$. Therefore, $h$ is not a physical measure. ∎

##  6\. Example

In this section, we present a nontrivial example to which our theory applies.
More precisely, we provide an example of an infinite dimensional system where
for very small coupling strength, the system admits a unique physical measure,
while for not so small coupling strength, the system admits at least three
physical measures, see Figure 1 for an illustration. We also discuss which
measures are physical. On the contrary, when considering the finite-
dimensional version of our example with the same coupling strength, we show
that the system admits a unique physical measure. Hence, the existence of
multiple physical states is a purely infinite dimensional phenomenon, i.e.,
the systems exhibit a phase transition.

Let $A\in\operatorname{SL}(2,{\mathbb{N}})$ be a hyperbolic matrix with
positive integer entries. Let $T_{0}(x)=Ax\mod 1$ the corresponding linear
automorphism with eigenvalues $\lambda,\lambda^{-1}$, $\lambda>1$. Let
$\rho\in\operatorname{Diff}^{\infty}({\mathbb{T}}^{2})$, and
$n_{*}\in{\mathbb{N}}$, then $T=\rho\circ T_{0}^{n_{*}}\circ\rho^{-1}$ is an
Anosov diffeormopfism of ${\mathbb{T}}^{2}$. If $J=\det(D\rho)$, then it can
be checked by an explicit computation that
$h_{*}=\left(J\circ\rho^{-1}\right)^{-1}$ is the SRB measure of $T$. Let
$\beta\in{\mathbb{R}}^{2}$, $\beta=(\sin\vartheta,\cos\vartheta)$,
$\vartheta\in(0,\pi)\cup(\pi,2\pi)$, and
$\frac{\beta_{1}}{\beta_{2}}\not\in{\mathbb{Q}}$.

For each $h\in{\mathcal{C}}^{\infty}({\mathbb{T}}^{2},{\mathbb{R}}_{+})$,
$\int h=1$, we define 121212 These are just special choices which allow to
exhibit an example for which we easily can perform explicit computation, many
other choices are as good.

$\begin{split}&\alpha(x)=1-\cos\langle\beta,x\rangle\\\
&\Phi_{\nu,h}(x)=x+\nu\beta\int\alpha h\\\ &T_{\nu,h}(x)=\Phi_{\nu,h}\circ
T(x)\quad\text{mod }1.\end{split}$ (6.1)

Also, we assume

$\rho(x)=\rho_{\mu}(x)=x+\mu\chi(x)\mod 1,$ (6.2)

where $\chi(x)=(\cos\langle\beta,x\rangle,\sin\langle\beta,x\rangle)$,
$\mu\in{\mathbb{R}}$, to be chosen later.

######  Lemma 6.1.

For each $n_{*}\in{\mathbb{N}}$, there exists $\mu_{*}>0$ such that for each
$\nu\geq 0$, $\mu\in[0,\mu_{*})$, and
$h\in{\mathcal{C}}^{\infty}({\mathbb{T}}^{2},{\mathbb{R}}_{+})$, $T_{\nu,h}$
is a Anosov diffeomorphism with uniform131313i.e., independent of $\nu$.
hyperbolicity.

###### Proof.

By defintion if $T_{\nu,h}(x)=T_{\nu,h}(y)$, then $T(x)=T(y)$ and hence $x=y$,
so $T_{\nu,h}$ is a diffeomorphism. On the other hand

$DT_{\nu,h}(x)=DT=D\rho_{\mu}\circ T^{n_{*}}_{0}\circ\rho_{\mu}^{-1}\cdot
DT^{n_{*}}_{0}\circ\rho_{\mu}^{-1}\cdot D\rho^{-1}.$

By construction, there exist two cones
${\mathcal{C}}_{1}\subset{\mathcal{C}}_{2}$, where the inclusion is strict,
such that $T^{n_{*}}_{0}{\mathcal{C}}_{2}\subset{\mathcal{C}}_{1}$ and, for
all $v\in{\mathcal{C}}_{2}$ $\|DT^{n_{*}}v\|\geq 2\|v\|$, provided $n_{*}$ is
large enough. Let ${\mathcal{C}}_{*}$ be a cone such that
${\mathcal{C}}_{2}\subset{\mathcal{C}}_{*}\subset{\mathcal{C}}_{1}$, where
again the inclusions are strict. We can then choose $\mu_{*}$ such that, for
all $\mu\in[0,\mu_{*})$,
$D\rho_{\mu}{\mathcal{C}}_{1}\subset{\mathcal{C}}_{*}$ and
$D\rho_{\mu}^{-1}{\mathcal{C}}_{*}\subset{\mathcal{C}}_{2}$. Accordingly,
$DT_{\nu,h}{\mathcal{C}}_{*}\subset{\mathcal{C}}_{*}$, strictly. In addition,
there exists $c>0$ such that, for $v\in{\mathcal{C}}_{*}$

$\|DT_{\nu,h}v\|\geq 2(1-c|\mu|)\|v\|$

hence, the vectors in the cone expand provided $\mu$ is small enough. The
hyperbolicity claim follows. ∎

Let ${\mathcal{B}}_{i}={\mathcal{B}}^{i,q-i}$, $i=0,\dots,3$,
$q-i\in{\mathbb{R}}_{+}$, $q\leq\bar{r}-2$, be the Banach spaces defined in
[14] using the cone ${\mathcal{C}}_{*}$.141414 Other Banach spaces, e.g. [2],
would work as well. We use [14] for convenience. We will also use the Banach
spaces
${\mathcal{B}}_{i}({\mathcal{C}}_{k})={\mathcal{B}}^{i,q-i}({\mathcal{C}}_{k})$
for the same construction using the cones ${\mathcal{C}}_{k}$. Note that,
${\mathcal{B}}_{i}({\mathcal{C}}_{2})\supset{\mathcal{B}}_{i}\supset{\mathcal{B}}_{i}({\mathcal{C}}_{1})$.

######  Lemma 6.2.

There exists $C>0$ and $\vartheta\in(0,1)$ such that, for all
$g\in{\mathcal{B}}_{i}$, $i\in\\{1,\dots,3\\}$,

$\|{\mathcal{L}}_{T_{\nu,h}}g\|_{{\mathcal{B}}_{i}}\leq\vartheta\|g\|_{{\mathcal{B}}_{i}}+C\|g\|_{{\mathcal{B}}_{i-1}}.$

###### Proof.

Note that

${\mathcal{L}}_{T_{\nu,h}}={\mathcal{L}}_{\Phi_{\nu,h}}{\mathcal{L}}_{\rho}{\mathcal{L}}_{T_{0}^{n_{*}}}{\mathcal{L}}_{\rho^{-1}}.$

By the definition of the Banach spaces it follows that, for $\mu$ small enough
and $n_{*}$ large, we have

$\begin{split}&\|{\mathcal{L}}_{\Phi_{\nu,h}}\|_{{\mathcal{B}}_{i}}\leq C\\\
&\|{\mathcal{L}}_{T^{n_{*}}}\|_{{\mathcal{B}}_{i}({\mathcal{C}}_{2})\to{\mathcal{B}}_{i}({\mathcal{C}}_{1})}\leq
1\\\
&\|{\mathcal{L}}_{\rho}\|_{{\mathcal{B}}_{i}({\mathcal{C}}_{1})\to{\mathcal{B}}_{i}}\leq
C\\\
&\|{\mathcal{L}}_{\rho^{-1}}\|_{{\mathcal{B}}_{i}\to{\mathcal{B}}_{i}({\mathcal{C}}_{2})}\leq
C.\\\ \end{split}$ (6.3)

Moreover, by Lemma 2.2 of [14] we have

$\|{\mathcal{L}}_{T_{0}^{n_{*}}}g\|_{{\mathcal{B}}_{i}({\mathcal{C}}_{1})}\leq
C\lambda^{-n_{*}\min\\{i,q-i\\}}\|g\|_{{\mathcal{B}}_{i}({\mathcal{C}}_{2})}+C\|g\|_{{\mathcal{B}}_{i-1}({\mathcal{C}}_{2})}.$
(6.4)

The lemma follows by (6.3) and (6.4). ∎

We verify assumptions (A2)-(A6) for the system defined in (6.1) using these
Banach spaces. Indeed, (A2) holds by definition of ${\mathcal{B}}_{i}$. For
assumption (A3), the fact that for any $a\in{\mathcal{C}}^{\bar{r}}$,
$\|ah\|_{{\mathcal{B}}_{i}}\leq
C_{\\#}\|a\|_{{\mathcal{C}}^{q}}\|h\|_{{\mathcal{B}}_{i}}$ follows as in [14,
Lemma 3.2], while the fact that $\left|\int h\right|\leq
C_{*}\|h\|_{{\mathcal{B}}_{0}}$ follows by definition of ${\mathcal{B}}_{0}$
(see [14, Section 4 ]). Also, (A4) is verified in [14, Lemma 2.1]. (A1) and
(A5) follow from Lemma 6.2 above. Notice ${\mathcal{L}}_{T_{\nu,h}}$ admits a
spectral gap on ${\mathcal{B}}_{i}$ for $i\in\\{1,2,3\\}$. This follows by
[14, Theorem 2.3], since $T_{\nu,h}$ is a toral Anosov map and hence
topologically transitive, see [17, Proposition 18.6.5]. Consequently,
${\mathcal{L}}_{T_{\nu,h}}$ admits a spectral decomposition; i.e., for any
$g\in{\mathcal{B}}_{i}$, $i\in\\{1,2,3\\}$,

${\mathcal{L}}_{T_{\nu,h}}g=h\int g+Q_{T_{\nu,h}}g.$ (6.5)

We delay the verification of assumption (A6) till later, namely after the
proof of Lemma 6.4 below. Before stating and proving the next lemma, we first
note that, (6.2) implies

$\rho_{\mu}^{-1}(x)=x-\mu\chi(x)+{\mathcal{O}}(\mu^{2}).$ (6.6)

Thus,

$\begin{split}&J\circ\rho_{\mu}^{-1}=e^{-\operatorname{Tr}\ln({\mathbbm{1}}+\mu
D\chi)\circ\rho_{\mu}^{-1}}=e^{-\operatorname{Tr}(\mu
D\chi-\frac{\mu^{2}}{2}(D\chi)^{2}+{\mathcal{O}}(\mu^{3}))\circ\rho_{\mu}^{-1}}\\\
&=\left[1-\mu\operatorname{Tr}(D\chi)+\frac{\mu^{2}}{2}\left[\operatorname{Tr}((D\chi)^{2})+[\operatorname{Tr}(D\chi)]^{2}\right]+{\mathcal{O}}(\mu^{3})\right]\circ\rho_{\mu}^{-1}\\\
&=1-\mu\operatorname{Tr}(D\chi)+\left[\operatorname{Tr}((D\chi)^{2})+[\operatorname{Tr}(D\chi)]^{2}+2\langle\nabla\operatorname{Tr}(D\chi),\chi\rangle\right]\frac{\mu^{2}}{2}\\\
&\phantom{=}+{\mathcal{O}}(\mu^{3})=1+a\mu+b\mu^{2}+{\mathcal{O}}(\mu^{3})\end{split}$
(6.7)

where

$\begin{split}a&=-\operatorname{Tr}(D\chi)=\beta_{1}\sin\langle\beta,x\rangle-\beta_{2}\cos\langle\beta,x\rangle\\\
b&=\operatorname{Tr}((D\chi)^{2})+[\operatorname{Tr}(D\chi)]^{2}+2\langle\nabla\operatorname{Tr}(D\chi),\chi\rangle\\\
&=(\beta_{1}^{2}-\beta_{2}^{2})\sin^{2}\langle\beta,x\rangle-(\beta_{1}^{2}-\beta_{2}^{2})\cos^{2}\langle\beta,x\rangle-\frac{3}{2}\beta_{1}\beta_{2}\sin(2\langle\beta,x\rangle).\end{split}$
(6.8)

######  Lemma 6.3.

There exists $\mu_{\star}>0$ such that for all $\mu\leq\mu_{\star}$,
$\Theta(\nu,h)\not\equiv 0$.

###### Proof.

Recalling (2.9), we have

$\Theta(\nu,h)=-({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\partial_{\beta}H(\nu,h).$
(6.9)

Thus, if $\Theta(\nu,h)\equiv 0$, then, seeing $H(\nu,h)$ as a distribution,
we have

$0=\int e^{i\langle k,x\rangle}\partial_{\beta}H(\nu,h)=-i\langle
k,\beta\rangle\int e^{i\langle k,x\rangle}H(\nu,h).$

Since $\beta$ is an irrational vector $\langle k,\beta\rangle\neq 0$ for all
$k\in{\mathbb{Z}}^{2}\setminus\\{0\\}$. It follows that $H(\nu,h)-1$ is zero
against the dense set $\\{e^{i\langle k,x\rangle}\\}_{k\in{\mathbb{Z}}^{2}}$,
hence $H(\nu,h)=1$ as a distribution, and hence as an element of
${\mathcal{B}}_{1}$. However, for each $\varphi\in{\mathcal{C}}^{0}$,
recalling that the SRB measure of $T$ is
$(J\circ\rho^{-1})^{-1}\operatorname{Leb}$, we can write, setting
$\eta_{h}:=\beta\int\alpha h$

$\begin{split}\int\varphi&=\int\varphi\circ
T_{\nu,h}=\int\varphi\left(T(x)+\nu\beta\int\alpha
h\right)\frac{J\circ\rho^{-1}(x)}{J\circ\rho^{-1}(x)}\\\
&=\int\varphi\left(x+\nu\eta_{h}\right)\frac{J\circ\rho^{-1}\circ
T^{-1}(x)}{J\circ\rho^{-1}(x)}=\int\varphi\left(x\right)\frac{J\circ\rho^{-1}\circ
T^{-1}(x-\nu\eta_{h})}{J\circ\rho^{-1}(x-\nu\eta_{h})}.\end{split}$

It follows that $J\circ\rho^{-1}$ must be an invariant function for $T$. But,
$T$ being ergodic, $J\circ\rho^{-1}$ must then be constant contradicting
equations (6.7) and (6.8). ∎

######  Lemma 6.4.

There exist $\mu_{\star},c_{+},c>0$ such that for each $\mu\leq\mu_{\star}$,
and $n_{*}\geq c_{+}\ln\mu^{-1}$, any continuous curve
$(\nu(\tau),h_{\nu(\tau)})$ in
${\mathbb{R}}\times({\mathcal{B}}_{2}\cap{\mathbb{W}})$ such that
${\mathcal{L}}_{T_{\nu,h_{\nu}}}h_{\nu}=h_{\nu}$, the equation

$\nu\int\alpha\Theta(\nu,h_{\nu})=1$ (6.10)

has no solutions for $\nu\leq\frac{4}{3\mu}$ and at least two solutions in
each interval $I$ such that $\nu(I)\subset(\frac{5}{3\mu},\frac{c}{\mu^{2}})$
and $\nu(I)$ has length larger than $2\pi+c_{+}$.

###### Proof.

For each $\mu>0$, choosing $c_{+}$ large enough, we have

$\left\|{\mathcal{L}}_{T_{0}^{n_{*}-1}}g-\int
g\right\|_{{\mathcal{B}}_{3}}\leq\mu^{3}\|g\|_{{\mathcal{B}}_{3}}.$ (6.11)

This implies that, for all smooth $\varphi$ and $g\in{\mathbb{W}}$,

$\begin{split}\int\varphi{\mathcal{L}}_{T_{\nu,h}}g&=\int
g(x)\varphi\circ\Phi_{\nu,h}\circ\rho_{\mu}\circ
T_{0}^{n_{*}}\circ\rho_{\mu}^{-1}(x)dx\\\ &=\int
g\circ\rho_{\mu}(y)J(y)\varphi\circ\Phi_{\nu,h}\circ\rho_{\mu}\circ
T_{0}^{n_{*}}(y)dy\\\
&=\int\left[{\mathcal{L}}_{T_{0}^{n_{*}-1}}[g\circ\rho_{\mu}J]-\int
g\circ\rho_{\mu}J\right]\varphi\circ\Phi_{\nu,h}\circ\rho_{\mu}\circ T_{0}\\\
&\phantom{=}+\int g\int\varphi\circ\Phi_{\nu,h}\circ\rho_{\mu}.\end{split}$

Since

$\int\varphi\circ\Phi_{\nu,h}\circ\rho_{\mu}=\int\frac{\varphi}{J\circ\rho_{\mu}^{-1}\circ\Phi_{-\nu,h}},$

we have

${\mathcal{L}}_{T_{\nu,h}}g=\frac{{\mathcal{L}}_{T_{0}}\left[{\mathcal{L}}_{T_{0}^{n_{*}-1}}[g\circ\rho_{\mu}J]-\int
g\circ\rho_{\mu}J\right]}{J\circ\rho_{\mu}^{-1}\circ\Phi_{-\nu,h}}+\frac{1}{J\circ\rho_{\mu}^{-1}\circ\Phi_{-\nu,h}}\int
g.$

The first term after the equality can be estimated using (6.11), yielding, for
$i\in\\{1,2,3\\}$,

$\left\|{\mathcal{L}}_{T_{\nu,h_{\nu}}}g-\frac{1}{J\circ\rho_{\mu}^{-1}\circ\Phi_{-\nu,h}}\int
g\right\|_{{\mathcal{B}}_{i}}\leq C\mu^{3}\|g\|_{{\mathcal{B}}_{i}}.$ (6.12)

Note that, for $\mu$ small
$g\circ\rho_{\mu}\in{\mathcal{B}}_{1}({\mathcal{C}}_{2})$ and so the estimate
follows remembering also (6.3). Thus, recalling (6.7),151515 Recall that
$h_{\nu}$ may not be a function but only a distribution, hence
${\mathcal{O}}$, is meant with respect to the norm of ${\mathcal{B}}_{2}$.

$\begin{split}&h_{\nu}={\mathcal{L}}_{T_{\nu,h_{\nu}}}h_{\nu}=\frac{1}{J\circ\rho_{\mu}^{-1}\circ\Phi_{-\nu,h_{\nu}}}+{\mathcal{O}}(\mu^{3}\|h_{\nu}\|_{{\mathcal{B}}_{2}})\\\
&\phantom{h_{\nu}}=1-\mu
a\circ\Phi_{-\nu,h_{\nu}}+\mu^{2}\left[\frac{a^{2}}{2}-b\right]\circ\Phi_{-\nu,h_{\nu}}+{\mathcal{O}}(\mu^{3}\|h_{\nu}\|_{{\mathcal{B}}_{2}})\\\
&\phantom{h_{\nu}}=1-\mu
a\circ\Phi_{-\nu,h_{\nu}}+\mu^{2}\left[\frac{a^{2}}{2}-b\right]\circ\Phi_{-\nu,h_{\nu}}+{\mathcal{O}}(\mu^{3}).\end{split}$
(6.13)

Moreover, by (6.12), for $g\in{\mathbb{V}}$, and $i\in\\{1,2,3\\}$,

$\begin{split}&\left\|Q_{\nu}g\right\|_{{\mathcal{B}}_{i}}=\left\|{\mathcal{L}}_{T_{\nu,h_{\nu}}}g-h_{\nu}\int
g\right\|_{{\mathcal{B}}_{i}}\\\
&\phantom{\left\|Q_{\nu}g\right\|_{{\mathcal{B}}_{2}}}=\left\|{\mathcal{L}}_{T_{\nu,h_{\nu}}}g-\frac{1}{J\circ\rho_{\mu}^{-1}\circ\Phi_{-\nu,h_{\nu}}}\int
g\right\|_{{\mathcal{B}}_{1}}\leq
C\mu^{3}\|g\|_{{\mathcal{B}}_{i}}.\end{split}$ (6.14)

We assume that $\mu_{\star}$ is so small that
$C\mu_{\star}^{3}\leq\frac{1}{2}$. Thus, setting $\omega=\int\alpha h_{\nu}$,
and using Werner formulae,

$\begin{split}&\omega=\int_{{\mathbb{T}}^{2}}\alpha(x)h_{\nu}(x)dx=\int\alpha\left\\{1-\mu
a\circ\Phi_{-\nu,h}\right\\}+{\mathcal{O}}(\mu^{2})\\\
&=1+\mu\int\cos\langle\beta,x+\nu\beta\omega\rangle\left(\beta_{1}\sin\langle\beta,x\rangle-\beta_{2}\cos\langle\beta,x\rangle\right)+{\mathcal{O}}(\mu^{2})\\\
&=1+\frac{\mu}{2}\int\beta_{1}\left[\sin\langle\beta,2x+\nu\beta\omega\rangle-\sin\langle\beta,\nu\beta\omega\rangle\right]\\\
&\phantom{=}-\frac{\mu}{2}\int\beta_{2}\left[\cos\langle\beta,2x+\nu\beta\omega\rangle+\cos\nu\omega\right]+{\mathcal{O}}(\mu^{2})\\\
&=1-\frac{\mu}{2}\left(\sin\vartheta\sin\nu\omega+\cos\vartheta\cos\nu\omega\right)+{\mathcal{O}}(\mu^{2})\\\
&=1-\frac{\mu}{2}\cos(\nu\omega-\vartheta)+{\mathcal{O}}(\mu^{2}).\end{split}$
(6.15)

This implies $\left|\omega-1\right|\leq C|\mu|$. Recalling (6.9), (6.14) and
(6.13),

$\begin{split}&\int\alpha\Theta(\nu,h_{\nu})=-\int\alpha({\mathbbm{1}}-Q_{\nu})^{-1}\partial_{\beta}h_{\nu}\\\
&=-\int\alpha\partial_{\beta}h_{\nu}+{\mathcal{O}}(\mu^{2})=\int\partial_{\beta}\alpha\cdot
h_{\nu}+{\mathcal{O}}(\mu^{2})\\\
&=-\mu\int\sin\langle\beta,x+\nu\beta\omega\rangle\left[\beta_{1}\sin\langle\beta,x\rangle-\beta_{2}\cos\langle\beta,x\rangle\right]+{\mathcal{O}}(\mu^{2})\\\
&=-\frac{\mu}{2}\left[\beta_{1}\cos\nu\omega-\beta_{2}\sin\nu\omega\right]+{\mathcal{O}}(\mu^{2})\\\
&=\frac{\mu}{2}\sin(\nu\omega-\vartheta)+{\mathcal{O}}(\mu^{2})\end{split}$
(6.16)

We can now conclude our study of equation (6.10).

$\begin{split}\Gamma(\nu):=&\nu\int\alpha\Theta(\nu,h_{\nu})\\\
=&\frac{\mu\nu}{2}\sin\left(\nu-\vartheta-\nu\frac{\mu}{2}\cos(\nu\omega-\vartheta)+{\mathcal{O}}(\nu\mu^{2})\right)+{\mathcal{O}}(\nu\mu^{2})\\\
=&\frac{\mu\nu}{2}\sin\left(\nu(1+{\mathcal{O}}(\mu))-\vartheta\right)+{\mathcal{O}}(\nu\mu^{2}).\end{split}$
(6.17)

Thus, for $\mu$ small enough and $\nu\leq\frac{4}{3\mu}$ we have
$|\Gamma(\nu)|<1$. On the other hand, if
$\nu\in(\frac{5}{3\mu},\frac{c}{\mu^{2}})$, for some constant $c>0$ small
enough, we have that the equation (6.10) must have at least two solutions each
time that $\nu$ varies more than $2\pi+c_{+}$.

∎

We now verify (A6). Recalling the spectral decomposition of
${\mathcal{L}}_{T_{\nu,h}}$ in (6.5), which holds for ${\mathcal{B}}_{i}$,
with $i\in\\{1,2,3\\}$, inequality (6.14) yields for $g\in{\mathcal{B}}_{i}$
and $\mu$ small,

$\|Q_{T_{\nu,h}}\|_{{\mathcal{B}}_{i}}\leq\sigma$

uniformly, for some $\sigma\in(0,1)$. Thus, if $\int g=0$,
$\|g\|_{{\mathcal{B}}_{i}}\leq 1$, then for all
$\\{S_{j}\\}_{j=1}^{\bar{n}}\subset{\mathcal{T}}$

$\|{\mathcal{L}}_{S_{\bar{n}}}\cdots{\mathcal{L}}_{S_{1}}g\|_{{\mathcal{B}}_{i}}=\|Q_{S_{\bar{n}}}\dots
Q_{S_{1}}g\|_{{\mathcal{B}}_{i}}\leq\sigma^{\bar{n}},$

which verifies (A6) and shows that the present model satisfies all our
assumptions.

######  Lemma 6.5.

If $h_{\nu}\in{\mathbb{W}}$ is such that
${\mathcal{L}}_{T_{\nu,h_{\nu}}}h_{\nu}=h_{\nu}$, and

$\nu\int\alpha\Theta(\nu,h_{\nu})=1,$

then, for $\nu\in(4\mu^{-1},c\mu^{-2})$, with $c$ small enough,

$\begin{split}&\operatorname{Range}(D_{h_{\nu}}F)\oplus
D_{\nu}F={\mathbb{V}}\\\ &\int\alpha
D^{2}_{h}F|_{\nu,h_{\nu}}(\Theta(\nu,h_{\nu}),\Theta(\nu,h_{\nu}))={\mathcal{O}}(\mu)\neq
0\\\ &\int\alpha h_{\nu}\neq 0.\end{split}$

###### Proof.

For any $g\in{\mathbb{V}}$, let

$\phi=\nu\Theta(\nu,h_{\nu})\cdot a\int\alpha h_{\nu}-g,$ (6.18)

with

$a=\frac{\int\alpha g}{\int\alpha h_{\nu}}.$

Notice that, by definition, $\alpha>0$ outside of a Lebesgue null set, hence,
for $\mu$ small enough, $\int\alpha h_{\nu}>0$, hence $a$ is well defined.
This proves the third statement of the Lemma. Note that, by the hypothesis and
the definition of $\phi$ it follows $\int\alpha\phi=0$. Recalling Lemma 3.6 we
have

$\begin{split}&D_{h}F|_{(\nu,h_{\nu})}\phi=\phi\\\
&D_{\nu}F|_{(\nu,h_{\nu})}=-\Theta(\nu,h_{\nu})\int\alpha h_{\nu}.\end{split}$

Consequently (6.18) can be written as

$g=-\nu aD_{\nu}F|_{(\nu,h_{\nu})}-D_{h}F|_{(\nu,h_{\nu})}\phi,$ (6.19)

which proves the first statement of the Lemma.

To prove the second statement we use Lemma 3.6. Since
${\mathcal{L}}_{T_{\nu,h_{\nu}}}h_{\nu}=h_{\nu}$,
$\nu\int\alpha\Theta(\nu,h_{\nu})=1$, we get

$\begin{split}&D^{2}_{h}F|_{\nu,h_{\nu}}(\Theta(\nu,h_{\nu}),\Theta(\nu,h_{\nu}))\\\
&=-\Big{\\{}2\sum_{i,j}\beta_{i}\beta_{j}\left({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}}\right)^{-1}\partial_{x_{i}}\left[{\mathcal{L}}_{T_{\nu,h_{\nu}}}({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\partial_{x_{j}}h_{\nu}\right]\\\
&+({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}})^{-1}\sum_{i,j}\beta_{i}\beta_{j}\partial_{x_{i}}\partial_{x_{j}}h_{\nu}\Big{\\}}.\\\
&=2\left({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}}\right)^{-1}\partial_{\beta}{\mathcal{L}}_{T_{\nu,h_{\nu}}}\Theta(\nu,h_{\nu})+\left({\mathbbm{1}}-{\mathcal{L}}_{T_{\nu,h}}\right)^{-1}\partial_{\beta}^{2}h_{\nu}\end{split}$
(6.20)

Therefore, by (6.20) and (6.14), (6.12), (6.13), and (6.15) we obtain

$\begin{split}&\int\alpha
D^{2}_{h}F|_{\nu,h_{\nu}}(\Theta(\nu,h_{\nu}),\Theta(\nu,h_{\nu}))\\\
&=\int\alpha\left[2\left({\mathbbm{1}}-Q_{T_{\nu,h}}\right)^{-1}\partial_{\beta}{\mathcal{L}}_{T_{\nu,h_{\nu}}}\Theta(\nu,h_{\nu})+\left({\mathbbm{1}}-Q_{T_{\nu,h}}\right)^{-1}\partial_{\beta}^{2}h_{\nu}\right]\\\
&=\int\alpha\left[2\partial_{\beta}{\mathcal{L}}_{T_{\nu,h_{\nu}}}\Theta(\nu,h_{\nu})+\partial_{\beta}^{2}h_{\nu}\right]+{\mathcal{O}}(\mu^{3})\\\
&=-2\int(\partial_{\beta}\alpha)Q_{T_{\nu,h_{\nu}}}\partial_{\beta}h_{\nu}+\int(\partial_{\beta}^{2}\alpha)h_{\nu}+{\mathcal{O}}(\mu^{3})\\\
&=-\mu\int\cos(\langle\beta,x\rangle\langle[\beta_{1}\sin\langle\beta,x\rangle-\nu\omega)-\beta_{2}\cos(\langle\beta,x\rangle-\nu\omega)]+{\mathcal{O}}(\mu^{3})\\\
&=\frac{\mu}{2}\beta_{1}\sin\nu\omega+\frac{\mu}{2}\beta_{2}\cos\nu\omega+{\mathcal{O}}(\mu^{3})\\\
&=\frac{\mu}{2}\cos(\nu\omega-\vartheta)+{\mathcal{O}}(\mu^{3}).\end{split}$

By equation (6.16) we have that $\nu\int\alpha\Theta(\nu,h_{\nu})=1$ when

$|\sin(\nu\omega-\vartheta)|=\frac{2}{\nu\mu}+{\mathcal{O}}(\mu)\leq\frac{3}{4}.$

Hence, the Lemma. ∎

######  Remark 6.6.

Lemma 6.5 implies that when
$\nu(\bar{\tau})\int\alpha\Theta(\nu(\bar{\tau}),h_{\nu(\bar{\tau})}=1$, then
the curve $(\nu(\tau),h_{\nu(\tau)})$ has a turning point at $\bar{\tau}$ in
the bifurcation diagram of the invariant measures, see figure 1.

######  Proposition 6.7.

Let $\mu,c>0$ be as in Lemma 6.4. For $\nu\leq\frac{4}{3}\mu^{-1}$ small
enough the system defined in (6.1) admits a unique physical measure. Moreover,
for each $\nu\in(2\mu^{-1},c\mu^{-2})$, but for finitely many values, the
system admits at least three invariant measures, only two of which are
physical measures.

###### Proof.

Lemma 4.2, Lemma 6.4, and Lemma 6.5 show that the system defined in (6.1)
satisfies the assumptions of Theorem 2.8. We can thus consider the curve
starting from the unique physical invariant measure at $\nu=0$ and follow it
till its first turning point. Note that, by Lemma 4.2, there can be only one
invariant measure, and hence no turning points, for all
$\nu\leq\frac{4}{3\mu}$. Hence, once we reach a turning point, the curve
cannot go back to $\frac{4}{3\mu}$ and must have another turning point. Let
$\nu_{1}$ be the first turning by Lemma 6.4 the curve must meet another
turning point $\nu_{2}\geq\nu_{1}-2\pi-c_{+}$. We can then keep following the
curve till the next turning point $\nu_{3}$ that, always by Lemma 6.4 must
occur before $\nu_{2}+2\pi+c_{+}$. Note that Theorem 2.8 shows that the curve
cannot backtrack in the interval $(\nu_{2},\nu_{1})$ infinitely many times,
hence the curve must keep moving toward increasing values of $\nu$ till it
exists the interval $(0,c\mu^{-2})$. The curve will have at least $C\mu^{-1}$
turning points $\nu_{i}$, for some constant $C>0$. Lemma 6.4 implies then that
for each $\nu\geq\frac{5}{3}\mu^{-1}$, $\nu\neq\nu_{i}$, there must at least
three invariant measures.

To verify which of the invariant measures are physical, we use Proposition 5.1
and Theorem 2.12. Recall that equation (6.14) implies that, for $\mu$ small
enough, the essential spectral radius of ${\mathcal{L}}_{T_{\nu,h_{\nu}}}$ is
smaller than $1/2$. We can compute as in (6.16) and obtain, for
$z\in{\mathbb{C}}$, $|z|>1/2$,

$\begin{split}&\Xi(z)=\int\alpha\Theta(z)=-\nu
z^{-1}\int\alpha({\mathbbm{1}}-z^{-1}{\mathcal{L}}_{T_{\nu,h_{\nu}}})^{-1}\partial_{\beta}h_{\nu}.\end{split}$
(6.21)

This allows to compute

$\begin{split}&\Xi(z)=\int\alpha\Theta(z)=-\nu
z^{-1}\int\alpha({\mathbbm{1}}-z^{-1}Q_{\nu})^{-1}\partial_{\beta}h_{\nu}\\\
&=-\nu z^{-1}\int\alpha\partial_{\beta}h_{\nu}+{\mathcal{O}}(\nu\mu^{3})=\nu
z^{-1}\int\partial_{\beta}\alpha\cdot h_{\nu}+{\mathcal{O}}(\mu^{2})\\\ &=-\nu
z^{-1}\mu\int\sin\langle\beta,x+\nu\beta\omega\rangle\left[\beta_{1}\sin\langle\beta,x\rangle-\beta_{2}\cos\langle\beta,x\rangle\right]+{\mathcal{O}}(\nu\mu^{3})\\\
&=-\nu\frac{\mu}{2z}\left[\beta_{1}\cos\nu\omega-\beta_{2}\sin\nu\omega\right]+{\mathcal{O}}(\nu\mu^{3})\\\
&=\nu\frac{\mu}{2z}\sin(\nu\omega-\vartheta)+{\mathcal{O}}(\nu\mu^{3}).\end{split}$
(6.22)

We are interested in computing the maximal eigenvalue of ${\mathcal{D}}_{h}$
near a turning point $\tau_{0}$, where $\Xi(1)=1$. For $\tau\neq\tau_{0}$,
Proposition 5.1 implies that 1 is an eigenvalue and the other eigenvalues of
modulus bigger than 1/2 are the solutions of the equation $\Xi(z)=1$. Note
that (6.22) implies that, for $|\nu-\nu_{0}|\leq c_{1}\mu$, $c_{1}$ small
enough, $\Xi(z)=1$ can have solutions only if
$z\in\Omega:=\\{z\in{\mathbb{C}}\;:\;|\Im(z)|\leq C_{\sharp}\nu_{0}\mu^{3}\leq
C_{\sharp}\mu;\;|\Re(z)-1|\leq cC_{\sharp}\\}$, for some constant
$C_{\sharp}>0$. If we consider $|\tau-\tau_{0}|\leq c_{2}\mu^{3}$, then
Lemmata 4.4 and 6.5 imply that $|\nu(\tau)-\nu_{0}|\leq c_{1}\mu$, provided
$c_{2}$ is chosen small enough.

Next, note that, recalling (6.22),

$\begin{split}\partial_{z}\Xi(z)&=\nu\int\alpha({\mathbbm{1}}z-{\mathcal{L}}_{T_{\nu(\tau),h_{\nu(\tau)}}})^{-2}\partial_{\beta}h_{\nu(\tau)}=z^{-2}\nu\int\alpha\partial_{\beta}h_{\nu(\tau)}+{\mathcal{O}}(\nu\mu^{3})\\\
&=-z^{-1}\Xi(z)+{\mathcal{O}}(\nu\mu^{3}).\end{split}$

Thus, for $\tau=\tau_{0}$ we have
$\partial_{z}\Xi(1)=-1+{\mathcal{O}}(\nu\mu^{3})$. On the other hand,

$\begin{split}&\partial_{\tau}\Xi(z)=-\nu^{\prime}z^{-1}\int\alpha({\mathbbm{1}}-z^{-1}{\mathcal{L}}_{T_{\nu,h_{\nu}}})^{-1}\partial_{\beta}h_{\nu}\\\
&-\nu
z^{-2}\int\alpha({\mathbbm{1}}-z^{-1}{\mathcal{L}}_{T_{\nu,h_{\nu}}})^{-1}\left[\partial_{h}{\mathcal{L}}_{T_{\nu,h_{\nu}}}\partial_{\tau}h_{\nu}+\partial_{\nu}{\mathcal{L}}_{T_{\nu,h_{\nu}}}\nu^{\prime}\right]\\\
&\times({\mathbbm{1}}-z^{-1}{\mathcal{L}}_{T_{\nu,h_{\nu}}})^{-1}\partial_{\beta}h_{\nu}\\\
&-\nu
z^{-1}\int\alpha({\mathbbm{1}}-z^{-1}{\mathcal{L}}_{T_{\nu,h_{\nu}}})^{-1}\partial_{\beta}\partial_{\tau}h_{\nu}.\end{split}$

Hence, for $\tau=\tau_{0}$, recalling Lemma 4.4, and equations (3.4), (4.14),

$\begin{split}\partial_{\tau}\Xi(z)|_{\tau=\tau_{0}}=&-\nu_{0}z^{-2}\int\alpha({\mathbbm{1}}-z^{-1}Q_{\nu_{0}})^{-1}\partial_{\beta}\left[{\mathcal{L}}_{T_{\nu,h_{\nu}}}({\mathbbm{1}}-z^{-1}Q_{\nu_{0}})^{-1}\partial_{\beta}h_{\nu}\right]\\\
&\times\int\alpha\Theta(\nu_{0},h_{\nu_{0}})+\nu_{0}z^{-1}\int\alpha({\mathbbm{1}}-z^{-1}Q_{\nu_{0}})^{-1}\partial_{\beta}\Theta(\nu_{0},h_{\nu_{0}})\\\
=&-z^{-2}\int\alpha({\mathbbm{1}}-z^{-1}Q_{\nu_{0}})^{-1}\partial_{\beta}\left[Q_{\nu_{0}}({\mathbbm{1}}-z^{-1}Q_{\nu_{0}})^{-1}\partial_{\beta}h_{\nu}\right]\\\
&+\nu_{0}z^{-1}\int\alpha({\mathbbm{1}}-z^{-1}Q_{\nu_{0}})^{-1}\partial_{\beta}\Theta(\nu_{0},h_{\nu_{0}}).\end{split}$

Recalling (6.14), (2.9), (6.13) and (6.8), we obtain

$\begin{split}\partial_{\tau}\Xi(z)|_{\tau=\tau_{0}}=&\nu_{0}z^{-1}\int\alpha\partial_{\beta}\Theta(\nu_{0},h_{\nu_{0}})+{\mathcal{O}}(\mu^{3})\\\
=&\nu_{0}z^{-1}\int[\partial_{\beta}^{2}\alpha]h_{\nu_{0}}+{\mathcal{O}}(\mu^{3})\\\
=&-\nu_{0}\mu z^{-1}\int\cos\langle\beta,x\rangle
a(x-\nu_{0}\beta\omega)+{\mathcal{O}}(\mu^{2}\nu_{0})\\\ =&-\nu_{0}\mu
z^{-1}\int\cos\langle\beta,x+\nu_{0}\beta\omega\rangle\left[\beta_{1}\sin\langle\beta,x\rangle-\beta_{2}\cos\langle\beta,x\rangle\right]+{\mathcal{O}}(\mu^{2}\nu_{0})\\\
&=-\frac{\nu_{0}}{2}\mu
z^{-1}\left[\beta_{1}\cos\nu_{0}\beta\omega-\beta_{2}\sin\nu_{0}\beta\omega\right]+{\mathcal{O}}(\mu^{2}\nu_{0})\\\
&=\frac{\nu_{0}}{2}\mu
z^{-1}\sin(\nu\omega-\vartheta)+{\mathcal{O}}(\mu^{2}\nu_{0})=\frac{\nu_{0}}{z\nu}\Xi(z)+{\mathcal{O}}(\mu^{2}\nu_{0}).\end{split}$

We can then apply the implicit function Theorem A.1 which, provided that $c$
is small enough, implies that in $\Omega$, all the solutions $z(\tau)$ of
$\Xi(z)=1$ satisfy

$\begin{split}z(\tau)&=1+z^{\prime}(\tau)(\tau-\tau_{0})+{\mathcal{O}}(\tau-\tau_{0})^{2}=1-\frac{\partial_{\tau}\Xi(1)}{\partial_{z}\Xi(1)}(\tau-\tau_{0})+{\mathcal{O}}(\tau-\tau_{0})^{2}\\\
&=1+\partial_{\tau}\Xi(1)(\tau-\tau_{0})+{\mathcal{O}}(\tau-\tau_{0})^{2}.\end{split}$

Accordingly, near the turning point, we have that the only eigenvalues outside
the disk of radius $\frac{1}{2}$ are $1$ and

$z(\tau)=1-\left\\{1+{\mathcal{O}}(\mu^{2}\nu_{0})\right\\}(\tau-\tau_{0})+{\mathcal{O}}(\tau-\tau_{0})^{2}.$

It follows that at a turning point $z$ crosses the unit circle, and hence,
only one branch can be physical. ∎

###  6.1. Phase transition

Phase transition in statistical mechanics usually means a situation where
finite size systems have a unique equilibrium state, while the corresponding
infinite system has more than one. It is then natural to enquire if the
multiple states that we have found in our example correspond to a phase
transition or not. In this subsection, we show that we are in the presence of
a real phase transition.
We start by recalling the discussion in [3] about the sense in which the
systems we are discussing in this paper are the limit of finite size systems.
Let $T_{0}=Ax\mod 1$, $A\in\operatorname{SL}(2,{\mathbb{N}})$, $T=\rho
T_{0}^{n_{*}}\rho^{-1}$ as in the previous subsections. Moreover, we choose
$\beta$ so that $T_{0}\beta=\lambda\beta$, $\lambda>1$. To simplify the
argument, we assume that $A^{*}=A$. Finally, we define
$T_{N,\nu}:{\mathbb{T}}^{2N}\to{\mathbb{T}}^{2N}$ as

$(T_{N,\nu}(x))_{i}=T(x_{i})+\frac{\nu}{N}\sum_{j=1}^{N}\beta\alpha(x_{j}).$

Suppose that $\mu_{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}$ converges
weakly to the measure $h(y)dy$. Then, for each
$\varphi\in\operatorname{Lip}({\mathbb{T}}^{d},{\mathbb{R}})$ we have

$\displaystyle\lim_{N\to\infty}\int\varphi\circ(T_{N,\nu}(x))\mu_{N}$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\varphi\left(T(x_{i})+\frac{\nu}{N}\sum_{j=1}^{N}\beta\alpha(x_{j})\right)$
$\displaystyle=\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}\varphi\left(T(x_{i})+\nu\beta\int\alpha
h\right)$ (6.23) $\displaystyle=\int\varphi\circ\left(T(x)+\nu\beta\int\alpha
h\right)h(x)dx$
$\displaystyle=\int\varphi{\mathcal{L}}_{T_{\nu,h}}h=\int\varphi\widetilde{\mathcal{L}}_{\nu}h.$

Thus, $\widetilde{\mathcal{L}}_{\nu}$ can be interpreted as the evolution of
the density of particles for the infinite system arising from the dynamics
$T_{N,\nu}$, see [18] for more details.
The question about phase transition can then be recast as:

  * •

how many physical measures does $({\mathbb{T}}^{dN},T_{N,\nu})$ have?

To answer, we need to understand better the map $T_{N,\nu}$.

######  Lemma 6.8.

There exist $\mu_{\star}<\frac{1}{2}$ such that, for each each
$N\in{\mathbb{N}}$, $|\mu|\leq\mu_{\star}$, $\nu\in{\mathbb{R}}$, the map
$T_{N,\nu}$ is Anosov.

###### Proof.

Let $W=\\{(\beta u_{1},\dots,\beta u_{N})\;:\;u_{i}\in{\mathbb{R}}\\}$. Let
$\pi$ be the orthogonal projection on $W$ and

${\mathcal{C}}_{+}=\\{v\in{\mathbb{R}}^{dN}\;:\;\|({\mathbbm{1}}-\pi)v\|\leq\sqrt{\mu}\|\pi
v\|\\}.$

Let $v\in{\mathcal{C}}_{+}$, then we can write $v=a+\sqrt{\mu}b$, where $a\in
W$ and $b\in W^{\perp}$. Then, recalling (6.6),

$\begin{split}(DT_{N,\nu}v)_{i}&=(D\rho
DT_{0}^{n_{*}}D\rho^{-1}(a+\sqrt{\mu}b))_{i}+\frac{\nu}{N}\sum_{j=1}^{N}\beta\langle\nabla\alpha(x_{j}),v_{j}\rangle\\\
&=(D\rho
DT_{0}^{n_{*}}[\tilde{a}+\sqrt{\mu}\tilde{b}])_{i}+\frac{\nu}{N}\sum_{j=1}^{N}\beta\langle\nabla\alpha(x_{j}),v_{j}\rangle\\\
&=(D\rho[\lambda^{n_{*}}\tilde{a}+\lambda^{-n_{*}}\sqrt{\mu}\tilde{b}])_{i}+\frac{\nu}{N}\sum_{j=1}^{N}\beta\langle\nabla\alpha(x_{j}),v_{j}\rangle\end{split}$

where $\tilde{a}\in W$, $\tilde{b}\in W^{\perp}$ with $\|a-\tilde{a}\|\leq
C\mu$ and $\|b-\tilde{b}\|\leq C\sqrt{\mu}$. Since, by the Cauchy-Schwarz
inequality,

$\left\|\left(\frac{1}{N}\sum_{j=1}^{N}\beta\langle\nabla\alpha(x_{j}),v_{j}\rangle\right)\right\|^{2}\leq
N^{-1}\left(\sum_{j=1}^{N}\|\nabla\alpha(x_{j})\|\|v_{j}\|\right)^{2}\leq
C\|v\|^{2},$

we get

$\left\|\pi\left[DT_{N,\nu}v\right]\right\|\geq\frac{\lambda^{n_{*}}}{2}\|v\|-C|\nu|\|v\|.$

Accordingly, for $|\nu|\leq c\mu^{-2}$ for $c$ small enough and choosing
$n_{*}$ such that $\lambda^{-n_{*}}\leq\mu^{3}$,

$\begin{split}&\left\|\pi\left[DT_{N,\nu}v\right]\right\|\geq\frac{\lambda^{n_{*}}}{4}\|v\|\\\
&\left\|({\mathbbm{1}}-\pi)\left[DT_{N,\nu}v\right]\right\|\leq\lambda^{-n_{*}}\|v\|+C_{\\#}\mu\lambda^{n_{*}}\|v\|\leq
C_{\\#}\mu\lambda^{n_{*}}\|v\|\\\
&\phantom{\left\|({\mathbbm{1}}-\pi)\left[DT_{N,\nu}v\right]\right\|}\leq
c_{\\#}C\mu\left\|\pi\left[DT_{N,\nu}v\right]\right\|\leq\sqrt{\mu}\left\|\pi\left[DT_{N,\nu}v\right]\right\|\end{split}$

for $\mu$ small enough. Therefore, ${\mathcal{C}}_{+}$ is an invariant cone
and, for each $v\in{\mathcal{C}}_{+}$,
$\|DT_{N,\nu}v\|\geq\frac{\mu^{-3}}{8}\|v\|$. Similar computations apply to
the cone

${\mathcal{C}}_{-}=\\{v\in{\mathbb{T}}^{dN}\;:\;\|\pi
v\|\leq\sqrt{\mu}\|({\mathbbm{1}}-\pi)v\|\\}$

and the map $T_{N,\nu}^{-1}$. Thus, $T_{N,\nu}$ is an Anosov map. ∎

Since the map $T_{N,\nu}$ is topologically mixing [17, Proposition 18.6.5], it
follows that the push forward of any absolutely continuous measure converges
to the SRB measure $\mu^{N}_{SRB}$, e.g. [14, Theorem 2.3]. In turn, this
implies that for Lebesgue almost all $x\in{\mathbb{T}}^{2N}$

$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\delta_{T_{N,\nu}^{k}(x)}=\mu^{N}_{SRB}.$
(6.24)

That is $\mu^{N}_{SRB}$ is the unique physical measure.

######  Remark 6.9.

Note that the definitions of physical measure for the finite and infinite
system differ substantially. However, they both have the fundamental character
in common: they refer to measures that are physically observable.

Let us connect the finite-dimensional case and the infinite-dimensional one.
For each $\varphi\in{\mathcal{C}}^{0}({\mathbb{T}}^{2},{\mathbb{R}})$,
defining $\pi_{i}:{\mathbb{T}}^{2N}\to{\mathbb{T}}^{2}$ as $\pi_{i}(x)=x_{i}$,
we can write

$\frac{1}{N}\sum_{i=1}^{N}\varphi(x_{i})=\left[\frac{1}{N}\sum_{i=1}^{N}\varphi\circ\pi_{i}\right](x).$

This means that if $x\in{\mathbb{T}}^{2N}$ is distributed according to an
exchangeable measure $\mu^{N}$ and we are interested in observables $\Phi_{N}$
of the form
$\Phi_{N}(x)=\left[\frac{1}{N}\sum_{i=1}^{N}\varphi\circ\pi_{i}\right](x)$,
$\varphi\in{\mathcal{C}}^{0}({\mathbb{T}}^{2},{\mathbb{R}})$, then

$\int\Phi_{N}(x)\mu^{N}(dx)=\int\varphi(y)(\pi_{1})_{*}\mu^{N}(dy)\\\ $

######  Lemma 6.10.

The SRB measure of $\mu^{N}_{SRB}$ is exchangeable.

###### Proof.

Equation (6.24) implies that, for Lebesgue almost all $x\in{\mathbb{T}}^{2N}$,

$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\left[\frac{1}{N}\sum_{i=1}^{N}\varphi\circ\pi_{i}\right](T^{n}_{N,\nu}(x))=\frac{1}{N}\sum_{i=1}^{N}\mu^{N}_{SRB}(\varphi\circ\pi_{i}).$

On the other hand if $\sigma:\\{1,\dots,N\\}\to\\{1,\dots,N\\}$ is a
permutation, then $\sigma\circ T_{N,\nu}=T_{N,\nu}\circ\sigma$, hence for each
$\psi,h\in{\mathcal{C}}^{0}({\mathbb{T}}^{2N},{\mathbb{R}})$ and $\int h=1$,

$\begin{split}\mu^{N}_{SRB}(\psi\circ\sigma)&=\lim_{n\to\infty}\int\psi\circ\sigma\circ
T_{N,\nu}^{n}h=\lim_{n\to\infty}\int\psi\circ
T_{N,\nu}^{n}h\circ\sigma^{-1}\\\ &=\mu^{N}_{SRB}(\psi)\int
h\circ\sigma^{-1}=\mu^{N}_{SRB}(\psi).\end{split}$

Thus, $\sigma_{*}\mu^{N}_{SRB}=\mu^{N}_{SRB}$, that is the measure
$\mu^{N}_{SRB}$ is exchangeable. ∎

Accordingly, $(\pi_{1})_{*}\mu^{N}_{SRB}$ is the unique measure that plays the
role of the physical invariant measure for the system of size $N$ and for the
relevant observables. It follows that we are witnessing a real phase
transition: the _non uniqueness_ of the physical invariant measure is an
infinite dimensional phenomenon.

##  Appendix A A quantitative Implicit Function Theorem

Since it is not readily available in the literature, in this appendix we
provide, for the reader’s convenience, a quantitative version of the implicit
function for functions between Banach spaces.

Let ${\mathcal{B}}$, and ${\mathcal{P}}$ be Banach spaces and let

$F\in{\mathcal{C}}^{0}({\mathcal{B}}\times{\mathcal{P}},{\mathcal{B}}).$

be Fréchet differentiable. Let
$(x_{0},\lambda_{0})\in{\mathcal{B}}\times{\mathcal{P}}$ such that
$F(x_{0},\lambda_{0})=0$. For each $\delta>0$ let
$V_{\delta}=\\{(x,\lambda)\in{\mathcal{B}}\times{\mathcal{P}}\\}\;:\;\|x-x_{0}\|_{{\mathcal{B}}}\leq\delta,\|\lambda-\lambda_{0}\|_{{\mathcal{P}}}\leq\delta\\}$.

######  Theorem A.1.

Assume that $F(x_{0},\lambda_{0})=0$, $\partial_{x}F(x_{0},\lambda_{0})$ is
invertible and assume there exists $\Upsilon,\delta>0$ such that

$\begin{split}&\sup_{(x,\lambda)\in
V_{\delta}}\|{\mathbbm{1}}-[\partial_{x}F(x_{0},\lambda_{0})]^{-1}\partial_{x}F(x,\lambda)\|_{{\mathcal{B}}}\leq\frac{1}{2}\\\
&\sup_{(x,\lambda)\in
V_{\delta}}\|[\partial_{x}F(x_{0},\lambda_{0})]^{-1}\partial_{\lambda}F(x,\lambda)\|_{{\mathcal{B}}}\leq\Upsilon.\end{split}$

Let
$\Delta:=\\{\lambda\in{\mathcal{P}}\>:\;\|\lambda-\lambda_{0}\|_{\mathcal{P}}\leq\frac{\delta}{2\Upsilon}\\}$.
Then there exists $g\in{\mathcal{C}}^{0}(\Delta,{\mathcal{B}})$, Fréchet
differentiable, such that all the solutions of the equation $F(x,\lambda)=0$
in the set
${\mathcal{K}}:=\left\\{(x,\lambda)\in{\mathcal{B}}\times{\mathcal{P}}\;:\;\|\lambda-\lambda_{0}\|_{\mathcal{P}}\leq\frac{\delta}{2\Upsilon},\|x-x_{0}\|_{{\mathcal{B}}}\leq\delta\right\\}$
are given by $\\{(g(\lambda),\lambda)\;:\;\lambda\in\Delta\\}$. In addition,

$\partial_{\lambda}g(\lambda)=-(\partial_{x}F(g(\lambda),\lambda))^{-1}\partial_{\lambda}F(g(\lambda),\lambda).$

###### Proof.

Let $A(x,\lambda)=\partial_{x}F(x,\lambda)$,
$M=\|A(x_{0},\lambda_{0})^{-1}\|$.

Let $\lambda$ be such that
$\|\lambda-\lambda_{0}\|_{\mathcal{P}}<\delta_{1}\leq\delta$. Consider
$U_{\delta}=\\{x\in{\mathcal{B}}\;:\;\|x-x_{0}\|_{{\mathcal{B}}}\leq\delta\\}$
and the function $\Theta_{\lambda}:U_{\delta}\to{\mathcal{B}}$ defined by

$\Theta_{\lambda}(x)=x-A(x_{0},\lambda_{0})^{-1}F(x,\lambda).$ (A.1)

If $F(x,\lambda)=0$ then $\Theta_{\lambda}(x)=x$, while if
$\Theta_{\lambda}(x)=x$, $F(x,\lambda)=0$. Next,

$\|\partial_{x}\Theta_{\lambda}\|=\|{\mathbbm{1}}-A(x_{0},\lambda_{0})^{-1}A(x,\lambda)\|\leq\frac{1}{2}$

for all $(x,\lambda)\in V_{\delta}$. So

$\|\Theta_{\lambda}(x_{0})-x_{0}\|=\|\Theta_{\lambda}(x_{0})-\Theta_{\lambda_{0}}(x_{0})\|\leq\left\|\int_{0}^{1}\partial_{\lambda}\Theta_{\xi}(x_{0})(\lambda-\lambda_{0})d\xi\right\|\leq\Upsilon\|\lambda-\lambda_{0}\|_{{\mathcal{P}}}.$

Thus, for $\|x-x_{0}\|\leq\delta$ and