Decay of correlations for unbounded observables. (English) Zbl 1477.37044
Summary: In this article, we study the decay rates of the correlation functions for a hyperbolic system \(T:M\rightarrow M\) with singularities that preserves a unique mixing SRB measure \(\mu\). We prove that, under some general assumptions, the correlations decay exponentially as \(n\rightarrow\infty\) for each pair of piecewise Hölder observables \(f,g\in L^p(\mu)\) and for each \(p>1\). As an application, we prove that the autocorrelations of the first return time functions decay exponentially for the induced maps of various billiard systems, which include the semi-dispersing billiards on a rectangle, billiards with cusps, and Bunimovich stadia. These estimates of the decay rates of autocorrelations of the first return time functions for the induced maps have an essential importance in the study of the statistical properties of nonuniformly hyperbolic systems (with singularities).
MSC:
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |
| 37C83 | Dynamical systems with singularities (billiards, etc.) |
| 37A50 | Dynamical systems and their relations with probability theory and stochastic processes |
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