Hitting time statistics for observations of dynamical systems

Abstract

In this paper we study the distribution of hitting and return times for observations of dynamical systems. We apply this results to get an exponential law for the distribution of hitting and return times for rapidly mixing random dynamical systems. In particular, it allows us to obtain an exponential law for random expanding maps, random circle maps expanding in average and randomly perturbed dynamical systems.

Matching strings in encoded sequences

“…(b) (Annealed decay of correlations) ∀n ∈ N * , ψ and φ Hölder observables from X to R, To prove this theorem, we will just apply Theorem 3.2 and Theorem 3.4 to the skew-product S with a well-chosen observation, following the idea given in [34].…”

“…The first example is a non-i.i.d. random dynamical system for which it was computed recurrence rates in [31] and hitting times statistics in [34].…”

“…In [36,31,34], the study of observed orbits (in particular, the study of return and hitting time) was used to obtain results for random dynamical systems. Following this idea, we combine ( ) with a particular observation f to obtain the following strong law of large numbers for random dynamical systems (provided that C ν exists)…”

“…Following the idea of [36,34] that the study of observation of dynamical systems can be used to study random dynamical systems, we will show in the next section that the previous result can be applied to obtain information on the shortest distance between two random orbits.…”

Decay of correlations and laws of rare events for transitive random maps

“…In [32], Rousseau studied hitting and return time statistics for observations of dynamical systems and got an annealed exponential law for super-polynomially mixing random dynamical systems. This theory was applied to random expanding maps, random circle maps expanding on average and randomly perturbed dynamical systems.…”