Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0143385711000897 How to cite this article: J.-R. CHAZOTTES and P. COLLET (2013). Poisson approximation for the number of visits to balls in non-uniformly hyperbolic dynamical systems.Abstract. We study the number of visits to balls B r (x), up to time t/µ(B r (x)), for a class of non-uniformly hyperbolic dynamical systems, where µ is the Sinai-Ruelle-Bowen measure. Outside a set of 'bad' centers x, we prove that this number is approximately Poissonnian with a controlled error term. In particular, when r → 0, we get convergence to the Poisson law for a set of centers of µ-measure one. Our theorem applies for instance to the Hénon attractor and, more generally, to systems modelled by a Young tower whose return-time function has an exponential tail and with one-dimensional unstable manifolds. Along the way, we prove an abstract Poisson approximation result of independent interest.

Extremes and Recurrence in Dynamical Systems

“…Condition (H3) (and versions thereof) have been stated and verified in [163,138,208] for the systems we consider here. …”

“…the 'u-Gibbs states' there. More recently, in [163], Hénon maps were considered (recall the simple model of Lozi maps above). Given real parameters a, b ∈ R, the Hénon map is the homeomorphism f a,b : R 2 → R 2 defined by f a,b (x, y) = (1−ax 2 +y, bx).…”

“…In [196] it is shown that there is a positive Lebesgue measure set of parameters (a, b) ∈ R 2 such that f a,b has an SRB measure µ a,b with good mixing properties (one should think of a being close to 2 and b being small here). In [163] it is shown that for µ a,b -typical points ζ, we have exponential HTS to balls around ζ. We do not give any details at this stage, but we remark that, in contrast to the uniformly hyperbolic toral automorphisms, the structure of the stable and unstable manifolds here is very complicated due to the criticality at (0, 0).…”

“…Examples include [163,138,81,208]. For these systems, rate estimates are possible in the spirit of Theorem 6.3.1 though there are complications in choosing the correct scaling u n := u/a n + b n as the invariant measure may fluctuate wildly.…”

“…This approach was first adopted in [72], and then applied more recently in [163,138,144,139,208,137]. We begin this chapter by reviewing this blocking argument approach.…”