Statistical properties of dynamical systems with some hyperbolicity.

*(English)*Zbl 0945.37009In this paper the existence and properties of invariant measures for a class of discrete dynamical systems with chaotic dynamics is studied. The studied systems include, apart from systems with hyperbolic attractors, piecewise hyperbolic maps such as Lozi maps, logistic maps, Hénon maps, and billiards with convex scatterers on a torus.

In a general set up, horseshoes with infinitely many branches and variable return times are introduced. These horseshoes occur as return maps on part of the state space. The first part of the paper consists of a study of ergodic properties of these horseshoes. Under some conditions, including distortion estimates and the requirement that the domain on which the return time equals \(n\) is exponentially small in \(n\), these horseshoes carry invariant measures. These invariant measures are studied using a tower construction called Markov extension. They yield the existence of absolutely continuous SRB measures for the original dynamical system. The paper moreover establishes that correlations decay exponentially fast. Central limit theorems are also derived. The set up and results are nicely presented in an introductory section.

In the second part of the paper, the general set up is applied to a number of dynamical systems, among which logistic maps and billiards with convex scatterers. These systems are not hyperbolic, but have a localized source of nonhyperbolicity. This allows for the construction of the infinitely branched horseshoes and to check the necessary assumptions. Especially the application to billiards is novel and interesting. Logistic maps had been treated in L.-S Young’s earlier paper [Commun. Math. Phys. 146, 123–138 (1992; Zbl 0760.58030)]. Proofs of the results on Hénon type attractors appeared in a joint work with M. Benedicks [Astérisque 261, 13–56 (2000; Zbl 1044.37013)]. Situations with slower than exponential decay of correlations are treated in L.-S. Young [Isr. J. Math. 110, 153–188 (1999; Zbl 0983.37005)].

In a general set up, horseshoes with infinitely many branches and variable return times are introduced. These horseshoes occur as return maps on part of the state space. The first part of the paper consists of a study of ergodic properties of these horseshoes. Under some conditions, including distortion estimates and the requirement that the domain on which the return time equals \(n\) is exponentially small in \(n\), these horseshoes carry invariant measures. These invariant measures are studied using a tower construction called Markov extension. They yield the existence of absolutely continuous SRB measures for the original dynamical system. The paper moreover establishes that correlations decay exponentially fast. Central limit theorems are also derived. The set up and results are nicely presented in an introductory section.

In the second part of the paper, the general set up is applied to a number of dynamical systems, among which logistic maps and billiards with convex scatterers. These systems are not hyperbolic, but have a localized source of nonhyperbolicity. This allows for the construction of the infinitely branched horseshoes and to check the necessary assumptions. Especially the application to billiards is novel and interesting. Logistic maps had been treated in L.-S Young’s earlier paper [Commun. Math. Phys. 146, 123–138 (1992; Zbl 0760.58030)]. Proofs of the results on Hénon type attractors appeared in a joint work with M. Benedicks [Astérisque 261, 13–56 (2000; Zbl 1044.37013)]. Situations with slower than exponential decay of correlations are treated in L.-S. Young [Isr. J. Math. 110, 153–188 (1999; Zbl 0983.37005)].

Reviewer: Ale Jan Homburg (Utrecht)

##### MSC:

37D05 | Dynamical systems with hyperbolic orbits and sets |

37A25 | Ergodicity, mixing, rates of mixing |

37A50 | Dynamical systems and their relations with probability theory and stochastic processes |

37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |