Globally coupled Anosov diffeomorphisms: statistical properties (Q6155614)

The authors study statistical properties of infinite systems of globally coupled Anosov diffeomorphisms arising as an appropriate limit of finitely many coupled Anosov maps. Let \(d\ge 2\) and consider a \(d\)-dimensional compact manifold \(M\). For \(r>1\) consider a transitive Anosov diffeomorphism \(T\in\mathrm{Diff}^r(M)\). Denote the \(N\)-fold products \(T\times\cdots\times T\) and \(M\times\cdots\times M\) by \(T_N\) and \(M_N\), respectively. Denote by \(\Phi^{\varepsilon}_N\) a diffeomorphism of \(M_N\) that is \(\varepsilon\)-close to \(\mathrm{Id}_{M^N}\). In particular, \(\Phi^0_N=\mathrm{Id}_{M^N}\). The system \((T_N\circ\Phi^{\varepsilon}_N,M_N)\) can be interpreted as a coupled system of \(N\) interacting units, where \(\Phi^{\varepsilon}_N\) accounts for the interaction between individuals, with strength tuned by the parameter \(\varepsilon\). Denote by \(\mathcal{M}_1(M)\) the set of probability measures over \(M\). Let \(\Psi_N:M_N\to\mathcal{M}_1(M)\) be the natural embedding given by \(\Psi_N(x)=\frac1N\sum_{i=1}^N\delta_{x_i}\). Let \(F_{\varepsilon}:M\times\mathcal{M}_1(M)\to M\) be \(C^r\), \(r>3\), in the first variable and continuous (with respect to the weak topology) in the second variable uniformly in \(x\). It is assumed that the coupling has the form \[ \bigl(\Phi_N^{\varepsilon}(x)\bigr)_i=F_{\varepsilon}\bigl(x_i,\Psi_N(x)\bigr), \] called \textit{mean field coupling}. For \(\mu\in\mathcal{M}_1(M)\) define \[ \Phi_{\mu}^{\varepsilon}=F_{\varepsilon}(\cdot,\mu). \] The map \[ (T\circ\Phi_{\mu}^{\varepsilon})_*:\mathcal{M}_1(M)\to\mathcal{M}_1(M) \] can be interpreted as the evolution of a state with infinitely many interacting units with state distribution given by \(\mu\). A measure \(h_{\varepsilon}\in\mathcal{M}_1(M)\) is called \textit{invariant} if \[ (T\circ\Phi_{h_{\varepsilon}}^{\varepsilon})_*h_{\varepsilon}=h_{\varepsilon}. \] Moreover, \(h_{\varepsilon}\) is called \textit{physical} if there is some \(h\in L^1\), such that for a fixed smooth volume form \(\omega\), the sequence \(\{\mu_n\}\) defined by \[ \mu_0=hd\omega\in\mathcal{M}_1(M),\qquad \mu_{n+1}=(T\circ\Phi_{\mu_n}^{\varepsilon})_*\mu_n \] converges weakly to \(h_{\varepsilon}\). The authors introduce two further technical assumptions on the coupling, (A1) and (A2) in the paper, which essentially impose that \(\Phi_h^{\varepsilon}\) is close to the identity, both in \(\varepsilon\) and \(h\), in an appropriate topology. Using transfer operators acting on anisotropic Banach spaces, the authors show that under the assumptions (A1) and (A2) there exists \(\varepsilon_0>0\) such that for all \(\varepsilon<\varepsilon_0\), the system admits a unique physical measure \(h_{\varepsilon}\). Moreover, they prove exponential convergence to equilibrium for a suitable class of distributions and show that the map \(\varepsilon\mapsto h_{\varepsilon}\) is Lipschitz continuous when \(|\varepsilon|\) is sufficiently small.