Quenched CLT for random toral automorphism

Ayyer, Arvind; Liverani, Carlangelo; Stenlund, Mikko

doi:10.3934/dcds.2009.24.331

Cited by 40 publications

(77 citation statements)

References 17 publications

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“…Here (2) states that, for typical sequences ω, the L 1 -distance between the push-forwards of two Hölder continuous densities tends to zero exponentially. The bound in (3) states that, with respect to any probability measure having a Hölder continous density, the random variables f and g • T ωn • · · · • T ω 1 become asymptotically decorrelated at an exponential rate.…”

Section: Resultsmentioning

confidence: 99%

A coupling approach to random circle maps expanding on the average

Stenlund

Sulku

2014

Stoch. Dyn.

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We study random circle maps that are expanding on the average. Uniform bounds on neither expansion nor distortion are required. We construct a coupling scheme, which leads to exponential convergence of measures (memory loss) and exponential mixing. Leveraging from the structure of the associated correlation estimates, we prove an almost sure invariance principle for vector-valued observables. The motivation for our paper is to explore these methods in a non-uniform random setting. models include [5,6,11,12,16,17,19]. After completing this paper, the authors have also learned of the very recent works [1,9] on the subject, as well as the related [28].The motivation for the paper is twofold. First, we wish to explore the suitability of the coupling method in the above context of non-uniform random maps. Diverting from the papers mentioned, the primary instrument in our analysis is indeed coupling. The coupling method is a soft tool for establishing statistical properties pertaining to the issues of memory loss and correlation decay. In the field of dynamical systems it has been implemented in various works such as [4,7,18,23,25,30] and many others. A transparent introduction to coupling for dynamical systems (in the most elementary setup) can be found in [27]. As to the second motivation, a question that arises naturally is whether other limit laws hold true; we wish to investigate the possibility of proving such laws for the present class of non-uniform random dynamical systems via correlation estimates. It was shown in [8, 20, 22] that a central limit theorem for Sinai billiards follows from correlation bounds involving suitable classes of observables. In [24] a similar approach was taken to prove an almost sure invariance principle (ASIP) for both random and non-random billiard systems. Here we show that, for our system, an ASIP follows from the established correlation estimates with little added work. Yet, the last point is subtle: it depends on the particular form of the correlation estimates, obtained for particular classes of observables. Let us be fully clear that the (averaged) theorems on the Markov chain corresponding to the random maps at issue can certainly be obtained, for example, via spectral methods. Here we present a different approach, which we hope to be of use to other authors beyond the present setup.The paper is organized as follows. In Sec. 2, we introduce the model precisely and record some mathematical preliminaries necessary for understanding the results and the proofs in the rest of the paper. In Sec. 3, we present our main results. In the following Secs. 4-7 we prove these results in the same order as they appear in Sec. 3. PreliminariesLet S denote the circle obtained by identifying the endpoints of the unit interval [0, 1]. The Lebesgue measure on S 1 is denoted by m.Given α ∈ (0, 1), we denote by C α the set of functions S → R (or S → C) that are Hölder continuous with exponent α. The corresponding Hölder constant is denoted by |f | α . We also introduce the norm

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Section: Resultsmentioning

confidence: 99%

A coupling approach to random circle maps expanding on the average

Stenlund

Sulku

2014

Stoch. Dyn.

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“…Furthermore, when 1 is simple (k 1 = 1), the existence of unit-length eigenvalues different from 1 is the only obstruction for the system to have the so-called mixing property 4 . In fact, if |γ 2 | < 1, then the exponential decay of correlations property holds: Let µ be the measure given by dµ dm = φ 1 .…”

Section: Transfer Operators and Quasi-compactnessmentioning

confidence: 99%

“…For a treatment of bifurcations in non-autonomous dynamical systems, we refer the reader to the monograph [72]. 4.4.1. Finite dimensional systems.…”

Section: Stabilitymentioning

confidence: 99%

Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems

González-Tokman¹

2018

Contemporary Mathematics

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“…where κ min = min(κ gray , κ white ). Note that b(x) is strictly positive by (1). In fact, there exists constants a min and b max such that…”

Section: 1mentioning

confidence: 99%

“…There exists ε > 0 such that the following holds. Let µ 1 be a probability measure on M, with a strictly positive, 1 6 -Hölder continuous density ρ 1 with respect to the measure µ. Given γ > 0, there exist 0 < θ γ < 1 and C γ > 0 such that…”

Section: Exponential Loss Of Memorymentioning

confidence: 99%

A Vector-Valued Almost Sure Invariance Principle for Sinai Billiards with Random Scatterers

Stenlund

2014

Commun. Math. Phys.

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Abstract. Understanding the statistical properties of the aperiodic planar Lorentz gas stands as a grand challenge in the theory of dynamical systems. Here we study a greatly simplified but related model, proposed by Arvind Ayyer and popularized by Joel Lebowitz, in which a scatterer configuration on the torus is randomly updated between collisions. Taking advantage of recent progress in the theory of time-dependent billiards on the one hand and in probability theory on the other, we prove a vector-valued almost sure invariance principle for the model. Notably, the configuration sequence can be weakly dependent and non-stationary. We provide an expression for the covariance matrix, which in the non-stationary case differs from the traditional one. We also obtain a new invariance principle for Sinai billiards (the case of fixed scatterers) with timedependent observables, and improve the accuracy and generality of existing results.

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Quenched CLT for random toral automorphism

Cited by 40 publications

References 17 publications

A coupling approach to random circle maps expanding on the average

A coupling approach to random circle maps expanding on the average

Multiplicative ergodic theorems for transfer operators: Towards the identification and analysis of coherent structures in non-autonomous dynamical systems

A Vector-Valued Almost Sure Invariance Principle for Sinai Billiards with Random Scatterers

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