Abstract
We study infinite systems of globally coupled Anosov diffeomorphisms with weak coupling strength. Using transfer operators acting on anisotropic Banach spaces, we prove that the coupled system admits a unique physical invariant state, \(h_\varepsilon \). Moreover, we prove exponential convergence to equilibrium for a suitable class of distributions and show that the map \(\varepsilon \mapsto h_\varepsilon \) is Lipschitz continuous.
Similar content being viewed by others
1 Introduction
Coupled systems are mathematical models of spatially extended systems consisting of identical interacting units. They provide a challenging subject of study from a mathematical point of view, give a well-motivated example of infinite dimensional dynamical systems, and often exhibit phase transition-like parameter-dependent behavior. Their popularity stems from the fact that they describe considerably well real-world systems (e.g. coupled oscillator networks [6, 26], heterogeneous networks [29] and networks with higher order interactions [7]).
In the field of dynamical systems coupled maps were introduced by Kaneko [22] and were first studied rigorously by Bunimovich and Sinai [10] in the case of nearest neighbhor interacting smooth expanding maps. The results of [10] were later extended to piecewise expanding maps [24, 25] and to coupled map lattices where the dynamics on each site is given by a smooth Anosov map [28].
An important type of coupled systems, which is different from the coupled map lattice model, is the globally coupled or mean field model. For example, gas particles interacting via their mean field give rise to the so-called Vlasov equation, [17, 32] or see [30, Chapter 5] for a simple derivation. In the case of plasma, a similar procedure gives the Vlasov-Poisson equation. Other examples of mean field models are the vorticity formulation of the two-dimensional Euler equation for incompressible fluids [13], and the time-dependent Hartree equation in quantum mechanics [21] (see [18] for more details). The study of the corresponding limiting equation is far from obvious as it may exhibit unexpected phenomena, e.g. Landau damping [27].
The above examples consider individual dynamics given by simple integrable motion. In the case in which the individual systems are strongly chaotic, statistical properties of the long-term behavior may be available. This was first shown by Keller [23] in the case of a toy model of globally coupled expanding maps. In this work, we study, for the first time, statistical properties of infinite systems of globally coupled Anosov diffeomorphisms that are motivated by considering an appropriate limit of finitely many coupled Anosov maps.
As already mentioned, the topic of infinitely many globally coupled maps was pioneered by Keller [23]. In [23] the local dynamics was described by an expanding circle map or a piecewise expanding map of the interval. For such coupled systems Keller [23] proved, in the case of weak coupling strength, the existence of a unique invariant state and exponential convergence to equilibrium (see also [8] for a similar result). In [5] a globally coupled system with site dynamics given by expanding fractional linear interval maps was studied. It was shown that the system undergoes a supercritical pitchfork bifurcation from a unique stable equilibrium to the coexistence of two stable and one unstable equilibrium. Both [5, 23] consider a coupling that only involves a parameter computed from the system state according to some fixed scheme. Later in [4] the work of [23] was extended to a more general coupling that mimics elastic interaction on the circle [14]. In addition to the analogous results of [23], Lipschitz continuity of the equilibrium state, as a function of the coupling strength parameter, was proved. This was taken one step further in [31] where linear response was shown in a rather general, smooth setting. Recent advances on globally coupled maps can be found in [15]. The work of [15] includes an abstract framework and applications to study statistical aspects of globally coupled circle maps. However, up to date there are no ergodic theoretic results on globally coupled higher dimensional hyperbolic systems in an infinite limit. This is because the right functional analytic tools to study hyperbolic systems, namely transfer operators acting on appropriate anisotropic Banach spaces, were not available until recently.
Starting with the paper [9], there has been a growing interest in developing anisotropic Banach spaces and spectral properties of transfer operators associated with hyperbolic dynamical systems. The books [2, 11] provide an extensive account of the topic.
Recently, a new family of anisotropic Banach spaces was introduced in [1] which are not only amenable to perturbations of the dynamics,Footnote 1 but for which the weak norms ‘behave’ like \(L^1\) and the strong norms ‘behave’ like BV, the space of functions of bounded variations ([1, Remark 2.15] for more details). These properties significantly simplify the study of the long-term behavior of the iterates of transfer operators associated with the globally coupled Anosov systems both for finite systems and in the mean field limit. Indeed, these spaces allow to define a simple invariant set of distributions under the action of the associated transfer operator (see (2.10)), where an invariant state of the system is proven to exist.
Yet, to prove finer statistical properties of globally coupled Anosov maps, we must introduce a higher order version of the spaces in [1] (see (2.6)). An important feature of this new space is that the transfer operator associated with the globally coupled system admits exponential memory loss with respect to its weak norm. Using this information we prove the uniqueness of the invariant state, \(h_\varepsilon \), exponential convergence to equilibrium and that the map \(\varepsilon \mapsto h_\varepsilon \) is Lipschitz continuous (see Theorem 2.9 below). This information allows proving that such measure is the unique physical measure of the system (see Theorem 2.3).
The paper is organised as follows. In Sect. 2 we introduce our system, state our main results (Theorems 2.3 and 2.9) and provide a strategy of the proof. In Sect. 3 we prove Theorem 2.9 in a series of lemmas and propositions. Theorem 2.3 is proved in the same section. Appendix A includes results on adapted foliations and test functions needed for the Banach spaces used in the paper. Appendix B includes statements and proofs about perturbations of Anosov maps. Appendix C includes properties of projections along the unstable direction, which are needed in the proof of the Lasota-Yorke inequality in Lemma 3.9.
2 The System and the Statement of the Main Result
2.1 The individual map
Let \(d \ge 2\) and consider a d-dimensional compact manifold M. Define the differentiable structure by the open cover \(\{V_i\}_{i=1}^S\) and charts \(\phi _i: V_i \rightarrow {{\mathbb {R}}}^d\), \(\phi _i \in {\mathcal {C}}^r\) for some \(r \ge 4\). More precisely, consider a fixed smooth partition of unity \(\{\vartheta _i\}\) subordinated to \(\{V_i\}_{i=1}^S\) and define a smooth volume form \(\omega \) by
All integrals will be understood with respect to such a form from now.
Consider an Anosov diffeomorphism \(T \in {{\,\textrm{Diff}\,}}^r(M)\), \(r>1\); i.e., there exists \(\lambda _0>1, \nu _0\in (0,1)\), \(c_0\in (0, 1)\) and a continuous cone field \({\mathcal {C}}=\{C(\xi )\}_{\xi \in M}\), \( \overline{C(\xi )}=C(\xi )\subset T_\xi M\) such that \(D_{\xi }T^{-1} C(\xi )\subset \text {int}(C(T^{-1}(\xi )))\cup \{0\}\) and
We will sometimes refer to it as the stable cone field (and the unstable cone field will be the complement.) We also assume that T is transitive.
2.2 Motivation for infinite coupled map systems
Denote the N-fold products \(T \times \cdots \times T\) and \(M\times \cdots \times M\) by \(T_N\) and \(M_N\), respectively. We view \((T_N,M_N)\) as a system of N units, (called either sites or particles in the coupled maps literature) each with a state in M evolving in time according to T. Define a diffeomorphism \(\Phi ^{\varepsilon }_N\) of the product manifold \(M_N\), \(\varepsilon \) close to \(Id_{M^N}\). We interpret \((T_N\circ \Phi ^{\varepsilon }_N, M_N)\) as a coupled system of N interacting units, where \(\Phi ^{\varepsilon }_N\) accounts for the interaction between individuals, with strength tuned by the parameter \(\varepsilon \)– in particular we assume that \(\Phi ^{0}_N=Id_{M^N}\).
The system state can be described by the vector \((x_1,\dots ,x_N)\), or equivalently, by the empirical measure \(\frac{1}{N}\sum _{i=1}^N \delta _{x_i}\). More precisely calling \({\mathcal {M}}_1(M)\) the set of probability measures over M, we can define the natural embedding \(\Psi _N: M_N\rightarrow {\mathcal {M}}_1(M)\) given by \(\Psi _N(x)=\frac{1}{N}\sum _{i=1}^N \delta _{x_i}\). Let
be \({\mathcal {C}}^r\), \(r>3\), in the first variable and continuous (with respect to the weak topology) in the second variable uniformly in x. We assume that the coupling has the form (mean field coupling)
For \(\mu \in {\mathcal {M}}_1(M)\), define
Note that, if \(\mu _N=\frac{1}{N}\sum _{i=1}^N \delta _{x_i}\), then
and, hence, in this case the dynamics on \(M_N\) induces a dynamics on \({\mathcal {M}}_1(M)\). Such dynamics extends naturally on all \({\mathcal {M}}_1(M)\). Thus, in the case in which \(\mu \) is a probability measure with a density, the map
can be interpreted as the evolution of a state with infinitely many interacting units with state distribution given by \(\mu \). Indeed, given a sequence of empirical measures \(\mu _N=\frac{1}{N}\sum _{i=1}^N \delta _{x_i}\) converging in the weak topology to some measure \(\mu \), as \(N \rightarrow \infty \), we have by hypothesis
where the convergence is in the uniform topology. Then, recalling (2.2), for each \(\varphi \in {\mathcal {C}}^0(M)\) we have
By (2.3), \(T\circ \Phi _{\varepsilon ,\mu _N}\rightarrow T\circ \Phi _{\varepsilon ,\mu }\) uniformly, so for each \(\delta >0\) there exists an \(N_\delta >0\) large enough such that, for all \(N\ge N_\delta \) it holds true
Thus we can write
This construction provides a motivation for the systems studied in the paper including the examples of Sect. 2.8.
2.3 Statement of the main results
Our aim is to study the long-time behavior of the self-consistent evolution \(\mu \mapsto (T\circ \Phi ^{\varepsilon }_{\mu })_*\mu \). In particular, we are interested in classifying the invariant measures and their stability properties.
Yet, the above setting and question are too general to allow a precise answer. We will thus introduce two further technical assumptionsFootnote 2 (A1), (A2) on the coupling that will be detailed in Sect. 2.5. As for the imprecision of the task, it arises from the possibility of having physically irrelevant invariant measures. For example, measures that represent a finite number of particles or that describe statistical properties we are not interested in. This problem appears already in the study of the invariant measures of an Anosov map and a typical solution is to restrict to physical measures. We, therefore, introduce an analogous definition for physical measures in the present situation of infinitely globally coupled systems.
Definition 2.1
We call a measure \(h_\varepsilon \in {\mathcal {M}}_1(M)\) invariant if
Moreover, we call an invariant measure \(h_\varepsilon \) physical if there exists some \(h\in L^1\), such that \(\mu _0=hd\omega \in {\mathcal {M}}_1(M)\), and, defining for each \(n\in {\mathbb {N}}\cup \{0\}\),
the sequence \(\{\mu _n\}\) converges weakly to \(h_\varepsilon \).
Remark 2.2
In essence, physical measures are measures that the system can asymptotically attain when starting with an initial condition that is absolutely continuous with respect to Lebesgue.
Our first main result is as follows.
Theorem 2.3
Under assumptions (A1), (A2) there exists \(\varepsilon _0>0\) such that, for all \(\varepsilon <\varepsilon _0\) the system admits a unique physical measure \(h_\varepsilon \).
Theorem 2.3 is the consequence of a more quantitative result, Theorem 2.9. To prove Theorem 2.9 we need to introduce a more suitable topology. This is done, in analogy with the strategy used for Anosov maps and flows, by introducing Banach spaces adapted to the dynamics.
2.4 Anisotropic BV
The following anisotropic Banach spaces, introduced in [1], will play a crucial role in this paper. To define these spaces, we need to consider appropriate foliations of M and test functions suited to such foliations. The spaces \(\Omega _{L,q,l}\) collect pairs \((W,\varphi )\) where W is a foliation, \(\varphi \) is a test function on M with controlled regularity on W, while the labels of the L, q, l are numbers: \(L>0\) is a uniform bound on some regularity class of the foliations, \(q\in {\mathbb {N}}\) is the number of derivatives we consider along the stable direction and l is the dimension of the target Euclidean space of \(\varphi \). For a precise definition see (A.10) in the Appendix A.2 and consult [1] for a detailed discussion. Given a function \(h\in {\mathcal {C}}^1(M,{\mathbb {C}})\) we define
for any \(q\in {\mathbb {N}}\cup \{0\}\) and some fixed \(a>0\). Let \({\mathcal {B}}^{0,q}\) be the Banach space obtained by completing \({\mathcal {C}}^1(M,{{\mathbb {R}}})\) in the \(\Vert \cdot \Vert _{0,q}\) norm. For each \(h\in {\mathcal {B}}^{0,q}\) let
We then define \({\mathcal {B}}^{1,q}:=\{h\in {\mathcal {B}}^{0,q}\;|\; \Vert h\Vert _{1,q}<\infty \}\).
Remark 2.4
According to [1, Lemma 2.12], there exists a canonical continuous injective map \(\iota :{\mathcal {B}}^{0,q}\rightarrow ({\mathcal {C}}^q)'\). In the following, we will use \(\iota \) to identify a positive element \(h\in {\mathcal {B}}^{0,q}\) with the measure \(h d\omega =\iota (h)\) without any further comment. The next lemma further clarifies this.
Lemma 2.5
A positive element of \({\mathcal {B}}^{0,q}\) is a measure. In addition, when restricted to \({\mathcal {M}}_1(M)\cap {\mathcal {B}}^{0,q}\), where \({\mathcal {M}}_1(M)\) denotes the set of probability measures over M, the norm \(\Vert \cdot \Vert _{0,q}\) is identical to \(\Vert \cdot \Vert _{TV}\), the total variation norm.
Proof
The first claim is standard as positive distributions are measures.Footnote 3 To conclude, note that, since \(\Vert \mu \Vert _{TV}=\sup _{\varphi \in {\mathcal {C}}^0}\int \varphi d\mu \), if \(d\mu = h d\omega \), we have \(\Vert h\Vert _{0,q}\le \Vert \mu \Vert _{TV}\). On the other hand
\(\square \)
For further use, we need to define a stronger norm, extending the spaces in [1]. For each \(h\in {\mathcal {C}}^2(M,{\mathbb {C}})\), define
Let \({\mathcal {B}}^{2,q}\) be the Banach space obtained by completing \({\mathcal {C}}^2(M,{{\mathbb {R}}})\) in the \(\Vert \cdot \Vert _{2,q}\) norm.
2.5 Assumptions on the coupling
We are now able to specify precisely our assumption on the coupling. Let us define \({\mathcal {B}}_1^{0,q}={\mathcal {M}}_1(M)\cap {\mathcal {B}}^{0,q}\).
We assume the coupling satisfies the following two conditions:
As already explained, we define the coupled map as
for \(h \in {\mathcal {B}}_1^{0,q}\) and \(\varepsilon \in {{\mathbb {R}}}\). This map represents the dynamics of a globally coupled map in the so-called thermodynamic limit with site dynamics \(T: M \rightarrow M\), system state given by the distribution h and coupling strength \(\varepsilon \).
Remark 2.6
Note that Eq. (A2) implies that, for each \(h \in {\mathcal {M}}_1(M)\), we have
Accordingly, there exists \(\varepsilon _0\) such that, for all \(\varepsilon \le \varepsilon _0\) the maps \(T_h^\varepsilon \) satisfy (2.1) uniformly with the same cones (see Lemma B.1 for details).
2.6 Transfer operators
We can now study the dynamics in the Banach spaces mentioned above. This is done by introducing a transfer operator acting on the anisotropic BV spaces. Recalling that these spaces can be canonically embedded into \(({\mathcal {C}}^{q+i})'\) (according to [1, Lemma 2.12], which can easily be extended to the case \(i=2\)), the transfer operator associated with T
can be defined as
Remark 2.7
Note that the precise version of (2.8) would be
As already remarked we allow the above imprecise notation (2.8) to simplify the notation. In particular, when \(h\in {\mathcal {C}}^1\), we identify h with the measure \(d\mu _h= h d\omega \) and the transfer operator associated to T is then given by
Clearly,
It follows that the evolution of the coupled system state is given by the self-consistent transfer operator \({\mathcal {L}}_{\varepsilon }: {\mathcal {B}}^{i,q}\cap {\mathcal {M}}_1(M) \rightarrow {\mathcal {M}}_1(M)\) that is defined as
where \({\mathcal {L}}_{T_h^{\varepsilon }}\) is the transfer operator associated with the map \(T_h^{\varepsilon }\) defined in (2.7). Indeed, for \(h\in L^1\)
Notice that
Observe that unlike \({\mathcal {L}}_{T_h^{\varepsilon }}\), the self-consistent transfer operator \({\mathcal {L}}_{\varepsilon }\) is nonlinear. Setting \(h_n={\mathcal {L}}_{\varepsilon }^n(h_0)\), if well defined, we can write
Note that if \({\mathcal {L}}_\varepsilon (h_\varepsilon )=h_\varepsilon \), then the coupled system admits an invariant state.
2.7 A more quantitative result
Our goal is to prove that the self-consistent transfer operator \({\mathcal {L}}_\varepsilon \) admits a unique fixed point and exhibits exponential convergence to equilibrium for a certain class of distributions. To do so we first define a compact convex subset of the \({\mathcal {B}}^{0,q+1}\) space. For \(K \ge 0\), define
and let \(\overline{{\mathcal {B}}}(K,q)\) be the closure of \({\mathcal {B}}(K,q)\) with respect to the \(\Vert \cdot \Vert _{0,q+1}\)-norm. The proof of the following proposition, which shows that \({\mathcal {L}}_\varepsilon \) can be iterated and that B(K, q) is eventually invariant, can be found in Sect. 3.2.
Proposition 2.8
There exists \(\varepsilon ^*_1>0\) such that, for all \(\varepsilon <\varepsilon ^*_1\) and \(q>0\),
There exists \(N \in {\mathbb {N}}\), and \(K_{\min } \ge 0\) such that, for all all \(|\varepsilon | < \varepsilon ^*_1\), \(n\ge N\) and \(K \ge K_{\min }\), \({\mathcal {L}}_{\varepsilon }^n (\overline{{\mathcal {B}}}(K,q)) \subset \overline{{\mathcal {B}}}(K,q)\).
The above is not enough to obtain uniqueness and exponential convergence. This is because, on the weak space \({\mathcal {B}}^{0,q}\), \({\mathcal {L}}_T\) does not admit a spectral gap. To overcome this hurdle we define a stronger Banach space than \({\mathcal {B}}^{1,q}\) and a set finer than \(\overline{{\mathcal {B}}}(K,q)\) to obtain our desired result.
Let \(K_1 \ge K_{\min }\), \(K_2 \ge 0\) and define
Let \(\overline{{\mathcal {B}}}(K_1,K_2,q)\) be the closure of \({\mathcal {B}}(K_1,K_2,q)\) with respect to the \(\Vert \cdot \Vert _{1,q+1}\)-norm.
Theorem 2.9
There exists \(\varepsilon ^*_2 > 0\), \(K_*(\varepsilon ^*_2) \ge 1\), such that for all \(q \in \{1,\dots ,r-3\}\), \(|\varepsilon | < \varepsilon ^*_2 \) and \(K_2\ge K_*(\varepsilon )\), the following holds:
-
(i)
\({\mathcal {L}}_\varepsilon \) has a unique fixed point \(h_\varepsilon \) in \(\overline{{\mathcal {B}}}(K_1,K_2,q)\). In addition, there exists \(\gamma \in (0,1)\) and \(C>0\) such that, for all \(h\in \overline{{\mathcal {B}}}(K_1,K_2,q)\), we have
$$\begin{aligned} \Vert {\mathcal {L}}_\varepsilon ^{n}(h)-h_\varepsilon \Vert _{1,q+1}\le C \gamma ^n . \end{aligned}$$
Moreover, for all \(|\varepsilon |, |\varepsilon '| < \varepsilon ^*_2 \),
-
(ii)
if \(h_\varepsilon \) and \(h_{\varepsilon '}\) are the unique fixed points of \({\mathcal {L}}_{\varepsilon }\) and \({\mathcal {L}}_{\varepsilon '}\) respectively, then there exists \(C > 0\) (depending on \(\varepsilon ^*_2\) and \(K_*\)) such that
$$\begin{aligned} \Vert h_\varepsilon -h_{\varepsilon '}\Vert _{1,q+1}\le C|\varepsilon -\varepsilon '|. \end{aligned}$$
The proof of the above Theorem is postponed to Sect. 3.4.
Remark 2.10
Item (ii) provides statistical stability of the coupled system. In particular, it implies that the map \(\varepsilon \mapsto h_\varepsilon \), \(|\varepsilon |\in [0,\varepsilon _2]\) is Lipschitz continuous.
The strategy to prove Theorem 2.9 is as follows: we first show that \({\mathcal {L}}_{\varepsilon }^N|_{\overline{{\mathcal {B}}}(K,q)}\) is continuous in the weak norm \(\Vert \cdot \Vert _{0,q+1}\) to conclude that \({\mathcal {L}}^N_\varepsilon \) has at least one fixed point in \(\overline{{\mathcal {B}}}(K,q)\). We then prove that \({\mathcal {L}}^N_\varepsilon \) is a contraction when acting on \(\overline{{\mathcal {B}}}(K_1,K_2,q)\), which gives the uniqueness and exponential convergence. We then prove that this fixed point is actually a unique fixed point of \({\mathcal {L}}_\varepsilon \) itself. Finally, we prove Lipschitz continuity of \(\varepsilon \mapsto h_\varepsilon \) by using the exponential convergence result and that \(\varepsilon \mapsto {\mathcal {L}}_\varepsilon \) is Lipschitz in a proper sense.
Note that the above strategy is natural when the transfer operator associated with the site dynamics admits a spectral gap on a Banach space. See [15] for a general strategy similar to the one we implement in this work.
Remark 2.11
If one wants to follow [15] in the Anosov setting, one has to choose the regularity in our spaces carefully. It seems to us that such a choice may then require more regularity on the map. Moreover, [15] assumes a ‘one step’ Lasota-Yorke inequality (see assumption (Con1) in [15] required to obtain the exponential convergence to equilibrium) which seems a strong assumption in a hyperbolic setting (the constant A that appears in our Lasota-Yorke inequality, does not only depend on the map, but also on the class of the foliations considered in our norms). Therefore, instead of verifying the assumptions of [15] which will force us to adding restrictive assumptions on T, we are going to pursue a different line of argument.
2.8 Examples
Before proving Theorem 2.3, Proposition 2.8, and Theorem 2.9, we provide a class of examples that satisfy assumptions (A1)–(A2). In both examples we consider \(M={\mathbb {T}}^d\) and \(\mu \in {\mathcal {M}}_1({\mathbb {T}}^d)\).
Example 2.12
Consider \(\Phi _{\mu }^{\varepsilon }\) given by the following formula:
for some \({\mathbb {K}}_1,{\mathbb {K}}_2 \in C^{\infty }({\mathbb {T}}^{d},{\mathbb {T}}^d)\).
Write this as
where
\(\gamma (x,\mu ({\mathbb {K}}_2)))={\mathbb {K}}_1(x) \cdot \mu ({\mathbb {K}}_2)\). Assume that \(\int d\mu (x)=\int h(x)dx\) for some \(h\in C^r({\mathbb {T}}^d)\). This is analogous to the one dimensional setting of [23].
Since
we have
and
Example 2.13
Now consider
for some \({\mathbb {K}}\in C^{\infty }({\mathbb {T}}^{2d},{\mathbb {T}}^d)\) (e.g. \({\mathbb {K}}(x,y)=\kappa (x-y)\) for diffusive coupling.) Then
Assumptions (A1)–(A2) are checked similarly:
and
3 Proofs
3.1 Transfer operators for sequential Anosov maps
The following Lasota-Yorke inequalities hold for the transfer operator \({\mathcal {L}}_T\):
Proposition 3.1
[1, Proposition 3.2, Lemma 4.1]. For each \(\theta \in (\max \{\nu _0,\lambda _0^{-1}\},1)\), there exist constants \(A, B>0\) such that, for all \(h\in {\mathcal {B}}^{0,q}\), \(q\in \{0,\dots ,r- 1\}\), holds true
In addition, for \(h\in {\mathcal {B}}^{1,q}\) and all \(q\in \{1,\dots ,r- 2\}\), holds true
Moreover, \(\{h\in {\mathcal {B}}^{1,q}\;:\;\Vert h\Vert _{1,q}\le 1\}\) is relatively compact in the topology associated to the norm \(\Vert h\Vert _{0,q+1}\).
Since we assumed that T is transitive, the above proposition implies that \({\mathcal {L}}_T\) admits a spectral gap when acting on \({\mathcal {B}}^{1,q}\), see [1] for a detailed discussion.
Remark 3.2
Choose \(\varepsilon ^* > 0 \) sufficiently small, and let \(h_0,\dots ,h_{n-1} \in {\mathcal {M}}_1(M)\). Each concatenation \(T_{h_{n-1}}^{\varepsilon }\circ \cdots \circ T_{h_0}^{\varepsilon }\) is a composition of Anosov diffeomorphisms with properties that can be uniformly controlled for any sequence \(h_i\), \(i=0,\dots ,n-1\), \(n \in {\mathbb {N}}\) and any \(|\varepsilon | < \varepsilon ^*\) (for an argument see Lemma B.1 in the Appendix). We will use this information to obtain uniform Lasota–Yorke inequalities for all \({\mathcal {L}}_{T_{h_{n-1}}^{\varepsilon }\circ \cdots \circ T_{h_0}^{\varepsilon }}\) provided \(|\varepsilon |\) is sufficiently small.
Proposition 3.3
There exists \(\varepsilon _1 > 0\) such that for each \(|\varepsilon | < \varepsilon _1\), \(\theta \in (\max \{\nu ,\lambda ^{-1}\},1)\) and constants \( A, B>0\) such that for all h, \(g_0,\ldots ,g_{n-1} \in {\mathcal {B}}_1^{0,q}\), \(n \in {\mathbb {N}}\), \(q\in \{0,\ldots ,r- 1\}\), holds true
In addition, for all h, \(g_0,\dots ,g_{n-1} \in {\mathcal {B}}^{1,q} \cap {\mathcal {B}}_1^{0,q}\) and all \(q\in \{1,\dots ,r- 2\}\), holds true
Proof
By Remark B.2, with small changes in notationFootnote 4 the proof follows verbatim as that of [1] Proposition 3.2]. \(\quad \square \)
The following statement is an essential perturbation lemma relating the action of the transfer operators to the distance of the associated coupled Anosov maps.
Lemma 3.4
Let \(h \in {\mathcal {B}}^{1,q} \cap {\mathcal {B}}_1^{0,q}\), \(q \in \{1,\dots ,r-2\}\), \(g, g_1, g_2 \in {\mathcal {M}}_1(M)\), \(|\varepsilon |, |\varepsilon '| < \varepsilon _2\) for some \(\varepsilon _2 \le \varepsilon _1\). Then
Proof
Assume \(|\varepsilon |\) is small enough, and the domains \(V_i\) of the charts are small enough such that there is an open set \(U_j\) for any i containing both \(T_{g_1}^{\varepsilon } V_i\) and \(T_{g_2}^{\varepsilon } V_i\), on which we can define functions \(\psi _j \in {\mathcal {C}}^r\), mapping \(U_j\) to \({{\mathbb {R}}}^d\). Then we can define \(T_t = tT_{g_1}^{\varepsilon }+(1-t)T_{g_2}^{\varepsilon }\), \(t \in [0,1]\), in the chart \(\{U_j, \psi _j\}\). The computation below should be understood in these charts.
According to Lemma B.3, \(T_t\) is an Anosov diffeomorphism such that the stable and unstable cones can be chosen uniformly not only in \(g_i\) and \(\varepsilon \) but also in t. We are going to write \(\varphi _t = \varphi \circ T_t\) for a test function \(\varphi \in {\mathcal {C}}^q(M,{\mathbb {C}}^{\ell })\) and \({\mathbb {F}}_t = T_t^{-1}{\mathbb {F}}\) for a foliation \({\mathbb {F}}\). Notice that the framework of [1] applies to this foliation, so in particular for \(T_t^{-1}W \in {\mathbb {F}}_t\) it holds that \(T_t^{-1}W \in {{\mathcal {F}}}^r_{\mathcal {C}}\). Let \(h \in {\mathcal {C}}^1(M,{\mathbb {C}})\) and \(\varphi \in {\mathcal {C}}^{q+1}(M,{\mathbb {C}})\).
First consider \(\Vert \varphi \circ T_t (T_{g_2}^{\varepsilon }-T_{g_1}^{\varepsilon })\Vert _{q+1}^{T_t^{-1}W}\) and for \(\xi \in M\) write \(({\mathbb {F}}_t)_\xi (x,y):=((F_t)_\xi (x,y),y)\) which describes the local foliation.Footnote 5
using [1, Lemma 2.15]. By Lemma A.1 we obtain
This gives
We can do a similar calculation for (II) and obtain
so
We obtained
and the fact that \({\mathcal {C}}^1(M,{\mathbb {C}})\) is dense in \({\mathcal {B}}^{0,q+1}\) concludes the proof (3.3).
For (3.4), an analogous argument works. We can prove an analogue of Lemma B.3 for \(T_t=tT_g^{\varepsilon }+(1-t)T_g^{\varepsilon '}\) (provided that \(|\varepsilon |, |\varepsilon '| < \varepsilon ^*\) of Lemma B.1) and repeat the above argument for \(T_g^{\varepsilon }, T_g^{\varepsilon '}\) instead of \(T_{g_1}^{\varepsilon },T_{g_2}^{\varepsilon }\). \(\quad \square \)
Using the assumptions on the coupling, we obtain the following corollary:
Corollary 3.5
By Lemma 3.4 and Assumption (A1) it follows
and by assumption (A2),
The above implies
and in particular for \(g_i,f_i \in \overline{{\mathcal {B}}}(K,q)\),
and by a similar argument
Next, we prove memory loss, which will also be needed for the proof of exponential convergence to equilibrium in the next subsection.
Lemma 3.6
Let \(K \ge K_{\min }\). There exists \(\varepsilon _3 > 0\), \(C > 0\) such that for all \(|\varepsilon | < \varepsilon _3\), \(q \in \{1,\dots ,r-2\}\)
holds true for all \(h\in {\mathcal {B}}^{1,q}\), \(h(1)=0\) and all \(g_1,\dots ,g_{n} \in {\mathcal {M}}_1(M)\).
Proof
By Proposition 3.3, Corollary 3.5 (in particular (3.8)) and the fact that \({\mathcal {L}}_T\) admits a spectral gap on \({\mathcal {B}}^{1,q}\), we have
for some constants \(B(m,K),B_1>0\) and \(\sigma \in (0,1)\), since \(h(1)=0\). Choose n so that \( A_1 \theta _0^n\le \frac{\theta }{3}\) for some \(\theta \in (0,1)\). Then choose m so that \(B_1\sigma ^m\le \frac{\theta }{3}\). Finally, choose \(|\varepsilon |\) small enough so that \(B(m,K)|\varepsilon |\le \frac{\theta }{3}\). Therefore,
This implies
\(\square \)
3.2 The invariance of \(\overline{{\mathcal {B}}}(K,q)\)
We show that \({\mathcal {L}}_\varepsilon \) is well defined and that \(\overline{{\mathcal {B}}}(K,q)\), defined in (2.10), is invariant.
Proof of Proposition 2.8
Let \(h\in {\mathcal {C}}^1(M)\). Note that for \(h\ge 0\), \({\mathcal {L}}_\varepsilon (h)\ge 0\) and \(\int {\mathcal {L}}_\varepsilon (h)=\int h\). Consequently, \({\mathcal {L}}_\varepsilon (h)\in {\mathcal {M}}_1(M)\cap {\mathcal {C}}^1(M)\). The first invariance results follow then by closing with respect to the \(\Vert \cdot \Vert _{q+1}\) norm and recalling Eq. (3.1).
Next, let \(h\in {\mathcal {B}}(K,q)\). By (3.1) we have
and by (3.2)
Thus, we have
Choose N large enough, such that \(A\theta ^N=\beta \in (0,1)\). By [1, Remark 2.15] we have \(\Vert h\Vert _{0,q+1} \le \Vert h\Vert _{1}\) (where \(\Vert h\Vert _{1}=\int _M |h|\)) and since we work with nonnegative distributions we have for each \(n\ge N\),
Choose K such that \(K \ge K_{\min }:=\frac{B}{1-\beta }\) holds. This completes the proof of the proposition. \(\quad \square \)
We are now ready to make a statment about the fixed points of \({\mathcal {L}}^N_\varepsilon \).
Proposition 3.7
There exists \(N \in {\mathbb {N}}\), \(K_{\min } \ge 0\), and \(\varepsilon _4 > 0\) such that \({\mathcal {L}}_{\varepsilon }^N\) has a fixed point in \(\overline{{\mathcal {B}}}(K,q)\) for all \(K \ge K_{\min }\) and all \(|\varepsilon | < \varepsilon _4\).
Proof
Let \(h_1,h_2 \in \overline{{\mathcal {B}}}(K,q)\). Note that, by equation (3.12) we have
Then, by Proposition 3.3 and Eq. (3.5) we have, for all \({\tilde{h}}_i\in {\mathcal {B}}(AK+B,q)\),
Hence, \({\mathcal {L}}^N_{\varepsilon }|_{\overline{{\mathcal {B}}}(K,q)}\) is continuous in the weak norm \(\Vert \cdot \Vert _{0,q+1}\), being the composition of continuous operators. Moreover, we have shown in Proposition 2.8 that \(\overline{{\mathcal {B}}}(K,q)\) is invariant under the action of \({\mathcal {L}}_\varepsilon ^N\). Since \(\overline{{\mathcal {B}}}(K,q)\) is a convex, compact metric space, we obtain that \({\mathcal {L}}^N_{\varepsilon }\) has at least one fixed point in \(\overline{{\mathcal {B}}}(K,q)\). \(\quad \square \)
Remark 3.8
Note that the Proposition 3.7 does not say much on the dynamics of \({\mathcal {L}}_\varepsilon \), just the existence of periodic orbits in \(\overline{{\mathcal {B}}}(K,q)\). To know if \({\mathcal {L}}_\varepsilon \) has a unique fixed point, some more work is needed.
3.3 Sequential Anosov maps and a stronger norm
In the following lemma we prove a Lasota-Yorke inequality for Anosov diffeomorphisms, for the Banach space \({\mathcal {B}}^{2,q}\) (see (2.6) to recall the definition of this space). In particular, Lemma 3.9 below can be useful outside the scope of this paper. We note however, that we do not prove that the unit ball of \({\mathcal {B}}^{2,q}\) is compactly embedded in \({\mathcal {B}}^{1,q+1}\) since the latter is not needed for the current work.Footnote 6
Lemma 3.9
There exists \(\theta \in (\max \{\nu ,\lambda ^{-1}\},1)\) and constants \( A_1, A_2>0\) such that for all \(h \in {\mathcal {B}}^{2,q}\), \(n \in {\mathbb {N}}\), \(q\in \{1,\dots ,r- 2\}\), holds true
Proof
All the operations in this proof are understood in the charts introduced in Sect. 2.1. We have
Accordingly, letting \((W,\varphi ^i)\in \Omega _{L,q+2, d}\) and using (3.15), we can write
where we have used the notation \( \sum _{i=1}^d\{DT^{-n}\}_{ji}\varphi ^i:= {\tilde{\varphi }}^j\). To continue, we need the following fact: for \(h,f,\varphi \in {\mathcal {C}}^r\),
Using (3.16) and integrating by part, we can write the above as
where \(\Vert \Theta ^1\Vert ^{T^{-n}W}_{q+1}+\Vert \Theta ^0\Vert ^{T^{-n}W}_{q+1}\le C_n\). This follows from the fact that \(T\in {\mathcal {C}}^r\) and [1, Lemma 2.15].
It remains to control the term with two derivatives. To this end we use the projectors \(\pi ^u,\pi ^s\) similarly to what is done in [1]. See appendix C for a precise defintion and their properties.
We can thus write
Let us analyze the above terms one by one. If we set
since \(\pi ^u\varvec{\eta }^j\) belongs to the unstable cone, using (C.2) twice we have, for the estimate of the first term
where \(\Vert {\bar{\varphi }}\Vert ^W_{q'}=\sup _{i,j}\Vert \varphi _i^j\Vert ^W_{q'}\) for each \(q'\le r\). Consequently, by [1, Lemma 2.11], we obtain
Note that the second and third terms in (3.18) are essentially the same. Indeed, exchanging i and j in the second yields the third apart from the fact that \(\varphi ^k_l\) is substituted by \(\varphi ^l_k\), which is irrelevant. We can thus analyze only the third term. By (C.3) and (C.2), we have
To treat the last term in (3.18), let
we can then write it as
Then, by (C.3) and (C.4) we have
The above implies that the last term is bounded by
By choosing \(\varpi \) large enough in (3.19), using (3.20) and (3.17), we get
The above equation can be iterated with steps \(n_0\) such that \(C\lambda ^{-2n_0}\le \theta <1\). Now choosing b in the definition of (2.6) large enough (depending on the fixed \(n_0\)) and using the Lasota-Yorke inequality in the \(\Vert \cdot \Vert _{1,q}\) from [1], we get for all \(n\in {\mathbb {N}}\)
\(\square \)
Using Remark 3.2 and Lemma 3.9, we obtain the following corollary:
Corollary 3.10
Let \(\varepsilon _1 > 0\) be the same as in Proposition 3.3. For each \(|\varepsilon | < \varepsilon _1\) and \(\theta \in (\max \{\nu ,\lambda ^{-1}\},1)\) there exist constants \( A_1, A_2>0\) such that for all h, \(g_0,\dots ,g_{n-1} \in {\mathcal {B}}^{2,q}\cap {\mathcal {B}}_1^{0,q}\), \(n \in {\mathbb {N}}\), \(q\in \{1,\dots ,r- 2\}\), holds true
that implies
Lemma 3.11
For \(h\in {\mathcal {B}}^{2,q}\cap {\mathcal {B}}_1^{0,q}\), \(q \in \{1,\dots ,r- 2\}\) we have
Proof
Recall the definition of \(T_t\) in the proof of Lemma 3.4. The calculation below is understood in the charts as explained at the beginning of the Proof of Lemma 3.4. Since
it is enough to estimate the latter since the former is covered by Lemma 3.4. We have
We first give a bound on (I).
We start with (I.I). Let \(D(T_{g_1}^{\varepsilon })^{-1}\circ (T_{g_1}^{\varepsilon })^{-1}\cdot \varphi ={\hat{\varphi }}\) and \(T_t = t \cdot T_{g_1}^{\varepsilon }+(1-t)\cdot T_{g_2}^{\varepsilon }, t \in [0,1]\). We have
Using what we obtained in the course of the Proof of Lemma 3.4, we get
As for (I.II), using Lemma A.2, we have
This implies
Combining (3.25) and (3.27) we obtain
We now bound (II).
For (II.I), by Lemma 3.4, we have
For (II.II), by Corollary A.3 we have
Using (3.28), (3.30) and (3.31) completes the proof of the first inequality of the lemma. The proof for the second part of the lemma follows similarly. \(\quad \square \)
Corollary 3.12
According to (A1), we can write
and similarly, according to (A2)
Furthermore,
where \(\theta \) is the contraction factor from Lemma 3.6. Similarly,
3.4 Fixed point uniqueness and exponential convergence in \({\mathcal {B}}(K_1,K_2,q)\)
Recall the definition of the set \({\mathcal {B}}(K_1,K_2,q)\) (see (2.11)).
Lemma 3.13
There exists \(N \in {\mathbb {N}}\), \(K_{1,\min } \ge K_{\min }\), \(K_{2,\min } \ge 0\) and \(\varepsilon _5 > 0\), for which \({\mathcal {L}}_{\varepsilon }^n({\mathcal {B}}(K_1,K_2,q))\subset B(K_1,K_2,q)\) and \({\mathcal {L}}_{\varepsilon }^n|_{{\mathcal {B}}(K_1,K_2,q)}\) is a strict contraction, with respect to the \(\Vert \cdot \Vert _{1,q+1}\) norm, for all \(K_i \ge K_{i, \min }\), \(|\varepsilon | < \varepsilon _5\), \(q \in \{1,\dots ,r-3\}\) and \(n\ge N\).
Proof
Let \(h\in {\mathcal {B}}(K_1,K_2,q)\), then by Proposition 2.8 we have \({\mathcal {L}}^n_\varepsilon (h)\subset {\mathcal {B}}(K_1,q+1)\). Next, by Corollary 3.10, we have
Choosing N large enough, such that \(A_1\theta ^N=\beta \in (0,1)\), and \(K_2\) such that \(K_2 \ge K_{2, \min }:=\frac{A_2K}{1-\beta }\), we have the wanted invariance.
Next, let \(h_1,h_2\in {\mathcal {B}}(K_1,K_2,q)\). Then for \(n\ge N\) we have \({\mathcal {L}}_\varepsilon ^n (h_1)={\mathcal {L}}_{T^\varepsilon _{h^{n-1}_1}}\dots {\mathcal {L}}_{T^\varepsilon _{h_1}}h_1\) and \({\mathcal {L}}_\varepsilon ^n (h_2)={\mathcal {L}}_{T^\varepsilon _{h^{n-1}_2}}\dots {\mathcal {L}}_{T^\varepsilon _{h_2}}h_2\) for some \(h_1,\dots ,h_1^{n-1}\) and \(h_2,\dots ,h_2^{n-1}\). Choose \(\gamma \in (\theta ,1)\) and \(C > \max \{1,C_1\}\). We prove by induction that \(\Vert {\mathcal {L}}_\varepsilon ^n (h_1)-{\mathcal {L}}_\varepsilon ^n (h_2)\Vert _{1,q+1} \le C\gamma ^n \Vert h_1-h_2\Vert _{1,q+1}\).
where for the first term we used (3.34) from Corollary 3.12, \(\Vert h_1\Vert _{2,q} \le K_2\) and the induction hypothesis, while for the second term we used Lemma 3.6.
Accordingly,
for \(|\varepsilon |\) sufficiently small. \(\quad \square \)
Proof of Theorem 2.9
We are now in a position to show that \({\mathcal {L}}_\varepsilon \) has a unique fixed point in \({\mathcal {B}}(K_1,K_2,q)\), for \(K_i\ge K_{i,\min }\). By Lemma 3.13 and the Banach fixed point theorem it follows that, for N large enough, \({\mathcal {L}}_\varepsilon ^N\) has a unique fixed point in \({\mathcal {B}}(K_1,K_2,q)\), call it \(h_\varepsilon \). Then
Accordingly, also \({\mathcal {L}}_\varepsilon h_\varepsilon \) is a fixed point of \({\mathcal {L}}_\varepsilon ^{N}\). On the other hand, by equations (3.14),(3.36) there exist constants \(A,B,A_1,A_2\) such that \({\mathcal {L}}_\varepsilon (h_\varepsilon )\in {\mathcal {B}}(K_1',K_2',q)\supset {\mathcal {B}}(K_1,K_2,q)\), where \(K_1'=AK_1+B\) and \(K_2'=A_1 K_2+A_2 K_1\). If N has been chosen large enough, \({\mathcal {L}}_\varepsilon ^N\) must have a unique fixed point in \({\mathcal {B}}(K_1',K_2',q)\) as well, which must be \(h_\varepsilon \). It follows \({\mathcal {L}}_\varepsilon (h_\varepsilon )=h_\varepsilon \). On the other hand, if \(g_\varepsilon \in {\mathcal {B}}(K_1,K_2,q)\) and \({\mathcal {L}}_\varepsilon (g_\varepsilon )=g_\varepsilon \), then \({\mathcal {L}}_\varepsilon ^N (g_\varepsilon )=g_\varepsilon \) and so, by unicity again, \(g_\varepsilon =h_\varepsilon \).
The proof of the first part of Theorem 2.9 is completed by (3.37) that implies
for all \(h \in {\bar{{\mathcal {B}}}}(K_1,K_2,q)\).
We now prove the second part of Theorem 2.9. Let \(h \in {\bar{{\mathcal {B}}}}(K_1,K_2,q)\). Then
Choose \(\lambda \in (0,1)\) and fix \(n^*\) such that \(C\theta ^{n^*} < \lambda \). Then
According to Corollary 3.12, for each \(n \in {\mathbb {N}}\) there exists \(C(n) > 0\) such that if \(h \in {\bar{{\mathcal {B}}}}(K_1,K_2,q)\),
We thus have
Let \(B(h,r) = \{g \in {\mathcal {B}}^{1,q}: \Vert g-h\Vert _{1,q} \le r \}\). Set \(r_0 = \frac{C(n^*)|\varepsilon -\varepsilon '|}{1-\lambda }\). Then \({\mathcal {L}}^{n^*}_{\varepsilon }B(h_{\varepsilon '},r_0) \subseteq B(h_{\varepsilon '},r_0)\). Indeed, let g be such that \(\Vert g-h_{\varepsilon '}\Vert _{1,q+1} \le r_0\). Then
According to the main theorem, we have \(\Vert {\mathcal {L}}^{kn^*}_{\varepsilon }(h)-h_{\varepsilon }\Vert _{1,q} \rightarrow 0\) for \(h \in \overline{{\mathcal {B}}}(K_1,K_2,q)\). Note that \(B(h_{\varepsilon '},r_0) \cap {\bar{{\mathcal {B}}}}(K_1,K_2,q) \ne \emptyset \). This implies that \(h_{\varepsilon } \in B(h_{\varepsilon '},r_0)\), i.e.
\(\square \)
3.5 Uniqueness of the physical measure
We can now conclude with the proof of Theorem 2.3.
Proof of Theorem 2.3
Let \(h_\varepsilon \) be a physical measure and let \(h\in L^1\) such that \({\mathcal {L}}_\varepsilon ^n (h)\) converges weakly to \(h_\varepsilon \). For each \(\delta >0\) we can find \(h_\delta \in {\mathcal {C}}^\infty \) such that \(\Vert h_\delta -h\Vert _{L^1}\le \delta \). Then, setting \(h_n={\mathcal {L}}_\varepsilon ^n (h)\), we have
It follows that, given any sequence \(\delta _n\rightarrow 0\), \({\mathcal {L}}_{T_{h_n}^\varepsilon }\cdots {\mathcal {L}}_{T_h^\varepsilon } h_{\delta _n}\) converges weakly to \(h_\varepsilon \). Moreover,
where \(C_\delta \rightarrow \infty \) when \(\delta \rightarrow 0\) and depends only on the fixed function h. We can thus choose a sequence \(\delta _n\) that goes to zero so slowly that \(C_{\delta _n}\theta ^n\rightarrow 0\) when \(n\rightarrow \infty \). Thus, for n large enough, \(\Vert {\mathcal {L}}_{T_{h_n}^\varepsilon }\cdots {\mathcal {L}}_{T_h^\varepsilon } h_{\delta _n}\Vert _{1,q}\le 2B\le K_1\). On the other hand, computing as above
The above implies that there exists a sequence \(\{g_n\}\subset {\mathcal {B}}(K_1,K_2,q)\cap {\mathcal {C}}^r\) such that \(g_n\) converges weakly to \(h_\varepsilon \).
Next, notice that \({\mathcal {L}}_{T_{h_\varepsilon }^\varepsilon }\) is the transfer operator associated with the Anosov map \(T_{h_\varepsilon }^\varepsilon \). Since \(T_{h_\varepsilon }^\varepsilon \) is a small perturbation of T, which is transitive, by structural stability \(T_{h_\varepsilon }^\varepsilon \) is transitive as well. It follows that \({\mathcal {L}}_{T_{h_\varepsilon }^\varepsilon }\), when acting on \({\mathcal {B}}^{1,q}\), has a spectral gap, see [1], and consequently \(T_{h_\varepsilon }^\varepsilon \) has a unique invariant measure in \({\mathcal {B}}^{1,q}\), call it \(g_*\). Therefore there exists constants \(C_*>0\) and \(\nu \in (0,1)\) such that, for each \(n\in {\mathbb {N}}\), we have
This implies that, for each \((W,\varphi )\in \Omega _{L,q,1}\), we have (recalling that \(h_\varepsilon ={\mathcal {L}}_\varepsilon (h_\varepsilon )={\mathcal {L}}_{T_{h_\varepsilon }^\varepsilon }h_\varepsilon \))
From the above, it follows
Taking the limit for \(n\rightarrow \infty \), since \(g_n\) converges weakly to \(h_\varepsilon \) we have
and taking the limit \(m\rightarrow \infty \) we have \(g_*=h_\varepsilon \) in \({\mathcal {B}}^{0,q}\), hence, recalling [1, Lemma 2.12], they are equal as distributions in \(({\mathcal {C}}^q)'\). But since \(g_*\) is a positive distribution they are both measures and hence they coincide as measures. It follows that \(h_\varepsilon =g_*\in \overline{{\mathcal {B}}}(K_1,K_2,q)\) and since the invariant measure in such a set is unique the theorem follows. \(\quad \square \)
Notes
See the earlier work of [3, 12, 19, 20] where Banach spaces that are amenable to perturbations of the dynamics were also constructed. See also [16] for Banach spaces amenable to perturbations, although limited to skew products. Similarly to the present case, the Banach spaces of [16] also resemble \(L^1\) and BV.
These assumption are essentially saying that \(\Phi ^\varepsilon _h\) is close to the identity, both in \(\varepsilon \) and h, in an appropriate topology.
If \(h\in ({\mathcal {C}}^{s})'\), \(s>0\), is positive, then for each \(\varphi \in {\mathcal {C}}^\infty \) we have \(h(\Vert \varphi \Vert _\infty \pm \varphi )\ge 0\), so \(|h (\varphi )|\le h(1)\Vert \varphi \Vert _\infty \) and the claim follows by the Riesz theorem.
Basically by replacing \(T^n\) in the proofs of [1] by \(T_{g_{n-1}}^{\varepsilon } \circ \cdots \circ {T_{g_0}^{\varepsilon }}\).
See subsection A.2 in the appendix for more details about foliations.
We warn the reader that to obtain information about the spectral properties of \({\mathcal {L}}_T\) when acting on \({\mathcal {B}}^{2,q}\), a compact embedding result is needed.
Here, and in the following, we use \(B_\delta (x)\) to designate \(\{z\in {\mathbb {R}}^{d'}\,\;: \Vert x-z\Vert \le \delta \}\) for any \(d'\in {\mathbb {N}}\).
Refer to Definition A.4 for the exact meaning of “connected component". Also note the abuse of notation since we use the same name for the sub-manifond in M and its image in the chart.
The fact that the intersection is non void and consists of exactly one point follows trivially from the fact that the foliation is adapted to the cone field, hence the two manifolds are transversal.
Note that in [1] the projectors where not explicitly defined.
References
Bahsoun, W., Liverani, C.: Anosov diffeomorphisms, anisotropic BV spaces and regularity of foliations. Ergodic Theory Dynam. Syst. 42(8), 2431–2467 (2022)
Baladi, V.: Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps. Springer International Publishing, Berlin (2018)
Baladi, V., Tsujii, M.: Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Annales de l’institut Fourier 57(1), 127–154 (2007)
Bálint, P., Keller, G., Sélley, F.M., Tóth, I.P.: Synchronization versus stability of the invariant distribution for a class of globally coupled maps. Nonlinearity 31(8), 3770–3793 (2018)
Bardet, J.B., Keller, G., Zweimüller, R.: Stochastically stable globally coupled maps with bistable thermodynamic limit. Commun. Math. Phys. 292(1), 237–270 (2009)
Bick, C., Timme, M., Paulikat, D., Rathlev, D., Ashwin, P.: Chaos in symmetric phase oscillator networks. Phys. Rev. Lett. 107(24), 244101 (2011)
Bick, C., Gross, E., Harrington, H.A., Schaub, M.T.: What are higher-order networks? (2021). arXiv preprint arXiv:2104.11329
Blank, M.L.: Self-consistent mappings and systems of interacting particles. Doklady Math. 83(1), 49–52 (2011)
Blank, M., Keller, G., Liverani, C.: Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15(6), 1905 (2002)
Bunimovich, L.A., Sinai, Y.G.: Spacetime chaos in coupled map lattices. Nonlinearity 1(4), 491 (1988)
Demers, M.F., Kiamari, N., Liverani, C.: Transfer operators in hyperbolic dynamics. An introduction., 33 Colloquio Brasilero de Matematica. Brazilian Mathematics Colloquiums series, Editora do IMPA. pp.252. (2021) ISBN 978-65-89124-26-9
Dyatlov, S., Zworski, M.: Stochastic stability of Pollicott–Ruelle resonances. Nonlinearity 28(10), 3511 (2015)
Euler, L.: Principes généraux du mouvement des fluides. Mémoires de l’académie des sciences de Berlin, 274–315 (1757)
Fernandez, B.: Breaking of ergodicity in expanding systems of globally coupled piecewise affine circle maps. J. Stat. Phys. 154(4), 999–1029 (2014)
Galatolo, S.: Self-consistent transfer operators: Invariant measures, convergence to equilibrium, linear response and control of the statistical properties. Commun. Math. Phys. 395(2), 715–772 (2022)
Galatolo, S., Lucena, R.: Spectral Gap and quantitative statistical stability for systems with contracting fibers and Lorenz like maps. Discrete Continuous Dyn. Syst. 40(3), 1309–1360 (2020)
Gärtner, J.: On the McKean–Vlasov limit for interacting diffusions. Math. Nachrichten 137(1), 197–248 (1988)
Golse, F.: On the dynamics of large particle systems in the mean-field limit. In: Muntean, A., Rademacher, J., Zagaris, A. (eds.) Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Lecture Notes in Applied Mathematics and Mechanics, vol. 3, pp. 1–144. Springer, Berlin (2016)
Gouëzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergodic Theory Dyn. Syst. 26(1), 189–217 (2006)
Gouëzel, S., Liverani, C.: Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties. J. Differ. Geom. 79(3), 433–477 (2008)
Hartree, D.R.: The wave mechanics of an atom with a non-Coulomb central field Part. I. Theory and methods. Math. Proc. Cambridge Philos. Soci. 24(1), 89–110 (1928)
Kaneko K.: (Ed.), Theory and Applications of Coupled Map Lattices, Wiley (1993)
Keller, G.: An ergodic theoretic approach to mean field coupled maps, pp. 183–208. Basel, Fractal geometry and stochastics II. Birkhäuser (2000)
Keller, G., Liverani, C.: Uniqueness of the SRB measure for piecewise expanding weakly coupled map lattices in any dimension. Commun. Math. Phys. 262(1), 33–50 (2006)
Keller, G., Liverani, C.: Map lattices coupled by collisions. Commun. Math. Phys. 291(2), 591–597 (2009)
Kuramoto, Y.: Chemical turbulence. In: Chemical Oscillations, Waves, and Turbulence, pp. 111–140. Springer, Berlin, Heidelberg (1984)
Mouhot, C., Villani, C.: On Landau damping. Acta Math. 207(1), 29–201 (2011)
Pesin Ya, B., Sinai Ya, G.: Space-time chaos in chains of weakly interacting hyperbolic mappings. Adv. Sov. Math 3, 165–98 (1991)
Pereira, T., van Strien, S., Tanzi, M.: Heterogeneously coupled maps: hub dynamics and emergence across connectivity layers. J. Eur. Math. Soc. 22(7), 2183–2252 (2020)
Spohn, H.: Large Scale Dynamics of Interacting Particles. Springer Science & Business Media, Berlin (2012)
Sélley, F.M., Tanzi, M.: Linear response for a family of self-consistent transfer operators. Commun. Math. Phys. 382(3), 1601–1624 (2021)
Vlasov, A.A.: The vibrational properties of an electron gas. Soviet Physics Uspekhi 10(6), 721 (1968)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no conflict of interest to disclose.
Additional information
Communicated by S. Dyatlov.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The research of W. Bahsoun is supported by EPSRC grant EP/V053493/1. C. Liverani acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006, and the INDAM-GNFM. The paper was partially supported by the Grant PRIN 2017S35EHN. W. Bahsoun would like to thank the hospitality of Leiden University, while C. Liverani and F. Sélley would like to thank the hospitality of Loughborough University during the course of this work.
Appendices
Appendix A. Test Functions and Foliations
1.1 A.1 Test functions
Let \({ \varpi }\ge 2\) be a parameter chosen as in [1, Equation (3.11)]. Denote by \(\alpha \) the multi-index \(\alpha =(\alpha _1,\cdots , \alpha _d)\) with \(\alpha _i\in {\mathbb {N}}\cup \{0\}\). Let \(|\alpha |=\sum _{i=1}^d\alpha _i\) and \(\partial ^{\alpha }=\partial _{x_1}^{\alpha _1}\cdots \partial _{x_d}^{\alpha _d}\). We thus define the weighted norm in \({\mathcal {C}}^\rho (M,{\mathcal {M}}(m,n))\), where \({\mathcal {M}}(m,n)\) is the set of the \(m\times n\) (possibly complex valued) matrices,
for some \({ \varpi }\ge 2\). Note that the above definition implies
The next lemma is Lemma 2.9 of [1].
Lemma A.1
For every \( \rho ,n, m,s \in {\mathbb {N}}\), \( \psi \in {\mathcal {C}}^\rho (M,{\mathcal {M}}(m,n))\) and \(\varphi \in {\mathcal {C}}^\rho (M,{\mathcal {M}}(m,s))\) we have
Moreover if \(\varphi \in {\mathcal {C}}^\rho (M,{\mathcal {M}}(m,n))\) and \(\psi \in {\mathcal {C}}^\rho (M,M)\), then
Lemma A.2
Let \(\varphi \in {\mathcal {C}}^{k}(M,{\mathbb {C}})\) and \(\psi ,{\tilde{\psi }} \in {\mathcal {C}}^k(M,M)\). Then
and
Proof
We are going to prove both formulas by induction. First recall that
We first prove (A.3). Using the above formula we compute
We now prove (A.4) by using (A.5).
For the second term we use the inductive assumption. This gives us the following:
\(\square \)
We obtain a useful corollary:
Corollary A.3
As a consequence of (A.3), we have
If furthermore \(\varphi \in {\mathcal {C}}^{k+1}(M,{\mathbb {C}})\), then by (A.4) we have
1.2 A.2 Foliations
To define the anisotropic spaces \({\mathcal {B}}^{0,q}\) and \({\mathcal {B}}^{1,q}\), we need to define a class of (stable) foliations adapted to the cone field, whose representation in local coordinates has certain uniform regularity. Let us recall be basic defintions from [1].
Definition A.4
A \({\mathcal {C}}^r\) t-dimensional foliation W is a collection \(\{W_\alpha \}_{\alpha \in A}\), for some set A, such that the \(W_\alpha \) are pairwise disjoint, \(\cup _{\alpha \in A}W_\alpha =M\) and for each \(\xi \in W_\alpha \) there exists a neighborhood \(B(\xi )\) such that the connected component of \(W_\alpha \cap B(\xi )\) containing \(\xi \), call it \(W(\xi )\), is a \({\mathcal {C}}^r\) t-dimensional open submanifold of M. We will call \({\mathcal {F}}^r\) the set of \({\mathcal {C}}^r\) \(d_s\)-dimensional foliations.
Definition A.5
A foliation W is adapted to the cone field \({\mathcal {C}}\) if, for each \(\xi \in M\), \(T_\xi W(\xi )\subset C(\xi )\). Let \({\mathcal {F}}^r_{\mathcal {C}}\) be the set of \({\mathcal {C}}^r\) \(d_s\)-dimensional foliations adapted to \({\mathcal {C}}\).
Given a \(d_s\)-foliation adapted to \({\mathcal {C}}\) we can associate to it local coordinates as follows. Let \(\delta _0>0\) be sufficiently small so that for each \(\xi \in M\) there exists a chart \((V_i,\phi _i)\) with \(\xi \in V_i\) and such that \(U_i:=\phi _i(V_i)\) contains the ball \(B_{\delta _0}(\phi _i(\xi ))\).Footnote 7 Also, choose \(U^0=U^0_u\times U^0_s\subset {\mathbb {R}}^{d_u}\times {\mathbb {R}}^{d_s}\) with \(U^0_u= B_{\delta _0/2}(0)\), \(U^0_s= B_{\delta _0/2}(0)\). Next, for each \(z\in U_i\), let W(z) be the connected component of \(\phi _i(W)\) containing z.Footnote 8 Define the function \(F_\xi :U^0\rightarrow {\mathbb {R}}^{d_u}\) by \(\{ (F_\xi (x,y)+x_\xi ,y_\xi +y)\}=\{(w,y+y_\xi )\}_{w\in {\mathbb {R}}^{d_u}}\cap W(x+x_\xi ,y_\xi )\), where \((x_\xi ,y_\xi )=\phi _i(\xi )\).Footnote 9 That is, \(W(x+x_\xi ,y_\xi )\) is exactly the graph of the function \(F_\xi (x,\cdot )+x_\xi \). Moreover,
In addition, we ask \(\delta _0\) to be small enough that the expression of DT in the above charts is roughly constant. See Lemma B.5 [1].
Now \({\mathbb {F}}_\xi (x,y)=(F_\xi (x,y),y)\), \((x,y) \in U^0\) describes the foliation locally. Denote by \({\mathbb {F}}\) the collection of maps \(\{{\mathbb {F}}_\xi \}\).
For each integer \(r\ge 2\) and \(L>0\), let
where
For each \(\varphi \in {\mathcal {C}}^{r}(M, {\mathbb {C}}^l)\) and \(W\in {\mathcal {F}}^r_{{\mathcal {C}}}\) let \(\varphi _{\xi ,x}(\cdot )=\varphi \circ \phi _{i}^{-1}\circ {\mathbb {F}}_{\xi }(x,\cdot )\), \(q\le r\) and define
We are finally able to define the sets \(\Omega _{L,q,l}\), \(L > 0\), \(r \ge 2\), \(q \in {\mathbb {N}}\cup \{0\}\) as
Appendix B. Some Properties of the Coupled Map \(T_h^{\varepsilon }\)
Lemma B.1
There exists \(\varepsilon ^* > 0\) such that
for all \(|\varepsilon |<\varepsilon ^*\) and \(h\in {\mathcal {B}}_1^{0,q}\); moreover there exists \(\lambda >1, \nu \in (0,1)\), \(c\in (0, 1)\) such that
for all \(|\varepsilon |<\varepsilon ^*\) and any sequence \(h_0,\dots ,h_{n-1}\in {\mathcal {B}}_1^{0,q}\), \(n \in {\mathbb {N}}\).
Proof
Fix \(\varepsilon ^* > 0\) such that (B.1) holds. Define \(\nu = (1-\varepsilon ^*)^{-1}\nu _0\) and \(\lambda = (1-\varepsilon ^*)\lambda _0\), where \(\nu _0\) and \(\lambda _0\) are given by (2.1). Decrease \(\varepsilon ^*\) further if necessary so that that \(0<\nu<1<\lambda \). Via a standard change of metric we may assume that in (2.1) \(c_0 = 1\). Note that for any h, \(|\varepsilon |<\varepsilon ^*\) and \(v\in C(\xi )\)
We then proceed by induction on n. Assume that for any \(v\in C(\xi )\)
Write \(T_{h_{n-2}}^{\varepsilon }\circ \cdots \circ T_{h_0}^{\varepsilon }=T^{\varepsilon }_{n-2}\). Then using the above two inequalities we obtain
Similarly, for \(v\notin C(\xi )\) we obtain
Finally, returning to the original metric accounts for the constant c in the statement of the lemma. \(\quad \square \)
Remark B.2
Lemma B.1 implies that for \(|\varepsilon |\) small enough, each \(T_h^\varepsilon \) is an Anosov diffeomorphism and any concatenation \(T_{h_{n-1}}^{\varepsilon }\circ \cdots \circ T_{h_0}^{\varepsilon }\) satisfies (2.1) with uniform constant c independent of \(\varepsilon \).
Lemma B.3
Let \(T_t=tT_{h_1}^\varepsilon +(1-t)T_{h_2}^\varepsilon \) for \(t \in [0,1]\) (understood in the charts defined in the proof of Lemma 3.4). There exists \(\varepsilon ^{**} > 0\) such that
for all \(|\varepsilon |<\varepsilon ^{**}\) and \(h_1,h_2\in {\mathcal {B}}_1^{0,q}\); moreover there exists \({\tilde{\lambda }}>1, {\tilde{\nu }}\in (0,1)\), \({\tilde{c}}\in (0, 1)\) such that
for all \(|\varepsilon |<\varepsilon ^{**}\), \(n \in {\mathbb {N}}\).
Proof
To simplify notation, let \(T_i=T_{h_i}^\varepsilon \). Write \(T_2^{-1} \circ T_1 = Id + \varepsilon S\). Then \(T_t = [Id + t \cdot \varepsilon S] \circ T_1\) and \(T^{-1}_t = T_1^{-1} \circ [Id + t \cdot \varepsilon S]^{-1}\).
We can see now that it is possible to fix \(\varepsilon ^{**} > 0\) such that (B.3) holds. Define \({\tilde{\nu }} = (1-\varepsilon ^{**})^{-1}\nu \) and \({\tilde{\lambda }} = (1-\varepsilon ^{**})\lambda \), where \(\nu \) and \(\lambda \) are given by (B.2). Decrease \(\varepsilon ^{**}\) further if necessary so that that \(0<{\tilde{\nu }}<1<{\tilde{\lambda }}\). Via a standard change of metric we may assume that in (2.1) \(c = 1\). Note that for any \(h_1,h_2\), \(|\varepsilon |<\varepsilon ^{**}\) and \(v\in C(\xi )\)
Similarly, for \(v\notin C(\xi )\) we obtain
Finally, returning to the original metric accounts for the constant c in the statement of the lemma. \(\quad \square \)
Appendix C. Projection Along the Unstable Direction
Here we follow [1] and introduce a way to project along the approximate stable and unstable directions. We do this by introducing projectors \(\pi ^u,\pi ^s\) which are only implicit in [1]. Note that the construction is local, so we can argue in one chart without further mentioning it. We start by recalling the construction in [1].
Consider the “almost unstable" foliation \(\Gamma =\{\gamma _s\}_{s\in {{\mathbb {R}}}^{d_s}}\) made of the leaves \(\gamma _s=\{(u,s)\}_{u\in {{\mathbb {R}}}^{d_u}}\) and its image \(T^n \Gamma \). The leaves of \(T^n \Gamma \) can be expressed in the form \(\{(x, {\tilde{G}}_n(x,y)\}\) for some function \({\tilde{G}}_n\), smooth in the x variable, with \(\Vert \partial _x {\tilde{G}}_n\Vert \le 1\) and the normalization \({\tilde{G}}_n(F(0,y),y)=y\). On the other hand, the leaves of W have the form \(\{(F(x,y),y)\}\). It is then natural to consider the change of variables \((x,y)=\Psi _n(x',y')\) where \((x,{\tilde{G}}_n(x,y'))=(F(x',y), y)\). Writing \(\varphi =(\varphi _1,\varphi _2)\), with \(\varphi _1\in {{\mathbb {R}}}^{d_u}\), \(\varphi _2\in {{\mathbb {R}}}^{d_s}\) we consider the decomposition defined in [1, Equations (3.5), (3.6)],Footnote 10
Where,
Let us check that \(\pi ^u,\pi ^s\) are indeed projectors. Note that
which immediately implies \((\pi ^u)^2=\pi ^u\). The computation for \(\pi ^s\) is similar.
The key properties of the above projectors are as follows.
By [1, Equation (3.10)] we have, for \((W,\varphi )\in \Omega _{L,q+2,d}\),
In addition, by [1, Equation (3.8)], we have
The second of [1, Equation (3.8)] implies also
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bahsoun, W., Liverani, C. & Sélley, F.M. Globally Coupled Anosov Diffeomorphisms: Statistical Properties. Commun. Math. Phys. 400, 1791–1822 (2023). https://doi.org/10.1007/s00220-022-04631-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-022-04631-3