Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities

Abstract

Dedicated to the memory of Nikolai Chernov.Abstract. We investigate a wide class of two-dimensional hyperbolic systems with singularities, and prove the almost sure invariance principle (ASIP) for the random process generated by sequences of dynamically Hölder observables. The observables could be unbounded, and the process may be non-stationary and need not have linearly growing variances. Our results apply to Anosov diffeomorphisms, Sinai dispersing billiards and their perturbations. The random processes under consideration are related to the fluctuation of Lyapunov exponents, the shrinking target problem, etc.2010 Mathematics Subject Classification. 37D50, 37A25, 60F17.

On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

“…However, when f is an unbounded observable, the CLT and ASIP may fail for some obvious reasons, for instance, f ∈ L 2 (µ) and thus the corresponding process {f • T n } n≥0 has no finite variance. In order to establish the limiting theorems for such process, we need some to add some extra conditions, such as moment controls in [11,12]. In this paper, we impose the following conditions on the dynamically Hölder series f ∈ H W,γ,t .…”

Quenched Normal Approximation for Random Sequences of Transformations

“…Quenched limit theorems for random dynamical systems are abundant in the literature, going back at least to Kifer [14]. Nevertheless they remain a lively topic of research to date: Recent central limit theorems and invariance principles in such a setting include Ayyer-Liverani-Stenlund [4], Nandori-Szasz-Varju [17], Aimino-Nicol-Vaienti [2], Abdelkader-Aimino [1], Nicol-Török-Vaienti [18], Dragičević et al [9,10], and Chen-Yang-Zhang [8]. Moreover, Bahsoun et al [5][6][7] establish important optimal quenched correlation bounds with applications to limit results, and Freitas-Freitas-Vaienti [11] establish interesting extreme value laws which have attracted plenty of attention during the past years.…”

Markov partition and Thermodynamic Formalism for Hyperbolic Systems with Singularities

“…for a sufficiently large N 0 ∈ N. By Lemma 5.3, x 0 = π(a) is a periodic point of ( R, F ) with period (2N 0 + 1), and thus a periodic point of (M, T ) with period (2N 0 τ 0 + τ 1 ). 11 Note that the points x 0 and F (x 0 ) = π(σ(a)) = x 0 both belong to R s a 0 ∩ R u a 0 ⊂ R. We denote the solid rectangles U s a 0 := U( R s a 0 ), U u a 0 := U( R u a 0 ), and U a 0 := U s a 0 ∩ U u a 0 , then the points x 0 and F (x 0 ) both belong to U a 0 .…”

“…The ASIP was first shown by Chernov [15] in the exponential case, and the vector-valued case was later proved by Melbourne and Nicol [39] using the martingale methods and then by Gouëzel [34] with the purely spectral methods. Under our assumptions (H1)-(H5), a non-stationary version of ASIP was established by the first and third authors in [11].…”