Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems (Q256223)

Let \((M,T)\) be a dynamical system on a compact metric space \(M\), which can be modeled by a Young tower. Suppose that the tail of the tower's return time function decays polynomially with degree greater than 7. Suppose also some regularity assumptions on the invariant (SRB) measure \(\mu\) admitted by the system.NEWLINENEWLINETo any ball \(B_\rho(x) \subset M\), associate the number of visits, say \(S^t_{\rho,x} (y)\), of the ball \(B_\rho(x)\) by the trajectory \(\{T^n(y)\mid 0\leq n < t/ \mu (B_\rho(x))\}\). For \(a,\rho > 0\), denote by \(\mathcal{N}_\rho(a)\) the set of \(x \in M\) such that \(B_\rho(x)\) has a return time smaller than \(a |\log\rho|\).NEWLINENEWLINEThen the main result asserts that there exist positive constants \(\kappa, a, C\) such that: for any sufficiently small positive radius \(\rho\) there exists \(\chi_\rho \subset M\) with \(\mu(\chi_\rho) < C |\log\rho|^{-\kappa}\), such that for all \(x \notin \chi_\rho \cup \mathcal{N}_\rho(a)\), \(y \in M\), \(k \in\mathbb{N}\) we have: NEWLINENEWLINE\[NEWLINE\Big|\mathbb{P}(S^t_{\rho,x}(y)=k)-e^{-t}\frac{t^k}{k!}\Big|\leq C|\log\rho|^{-\kappa}.NEWLINE\]NEWLINE NEWLINESupposing moreover that \(M\) is a manifold, \(T\) is a \(C^2\)-diffeomorphism and \(\lambda > 9\), the authors also prove that \(\mu(\mathcal{N}_\rho(a)) \leq C |\log\rho|^{-\kappa}\), too.NEWLINENEWLINEThus, up to error terms of order \(O(|\log\rho|^{-\kappa})\), the statistics of return times to a metric ball of radius \(\rho\) are asymptotically Poisson distributed. This applies to non-uniformly hyperbolic maps and to weakly regular invariant measures, for example SRB mesures on attractors.