Limit theorems for sequential expanding dynamical systems on [0,1]

Abstract

To cite this version:Jean-Pierre Conze, Albert Raugi. Limit theorems for sequential expanding dynamical systems on [0,1]. Contemporary mathematics, American Mathematical Society, 2007, 430, pp.89-121. hal-00365220 Limit theorems for sequential expanding dynamical systems on [0, 1] Jean-Pierre Conze and Albert RaugiAbstract. We consider the asymptotic behaviour of a sequence (θn), θn = τn • τ n−1 · · · • τ 1 , where (τn) n≥1 are non-singular transformations on a probability space. After briefly discussing some definitions and problems in this general framework, we consider the case of piecewise expanding transformations of the interval. Exactness and statistical properties (a central limit theorem for BV functions after a moving centering) can be shown for some families of such transformations.The method relies on an extension of the spectral theory of transfer operators to the case of a sequence of transfer operators.

Extremes and Recurrence in Dynamical Systems

“…Notice that the process X k (·) = φ(f ω k • · · · • f ω 1 (·)), equipped with the probability P given by the Lebesgue measure Leb is not necessarily stationary nor independent. By using the theory of sequential β-transformations developed in [254], we can apply the generalisation of Extreme Value Theory to non-stationary sequences obtained in the first part of [253] and actually verify the adapted conditions Д q (u n ) and Д q (u n ). We can also obtain that, when ζ is periodic, explicit expressions for the EI.…”

Statistical properties for compositions of standard maps with increasing coefficient

“…That being said, the statistical properties explored in this paper are closest to those on memory loss for nonautonomous compositions of hyperbolic maps [4,5,27] (see also [3]); Sinai billiards systems with slowly moving scatterers [10,28,29]; and polynomial loss of memory for intermittent-type maps of the interval with a neutral fixed-point at the origin [1,23]. We have benefited especially from the techniques in [13], which studies statistical properties of sequential piecewise expanding compositions in one dimension.…”

“…Thus, for Theorem B, it suffices to prove convergence in distribution of 1 √ N S M,N (X); for this, we approximate S M,N by a sum of Martingale differences with respect to the increasing filtrationŝ G M,k , k ≥ M . Alternatively, following the analogue of the derivation of a reverse Martingale difference approximation given in [13] for forward martingale differences, one can look for a Martingale differ-…”

Central limit theorems with a rate of convergence for time-dependent intermittent maps

“…Statistical properties of time-dependent dynamical systems have been studied with increasing focus over the past decade; see for instance [4, 5, 16, 18, 25-27, 33, 36, 41, 42, 44, 45,47]. Self-norming CLTs in the spirit of (1) for non-random compositions were obtained in [9,10] for a class of nearby hyperbolic maps, and in [12,20,34] for one-dimensional piecewise-expanding maps, where [20] established also rates of convergence with respect to the Kolmogorov metric. The general operator-theoretic approach of [12], which coarsely speaking applies to compositions of maps with quasicompact transfer operators on a suitable Banach, was used in [19] to show almost sure invariance principles also for higher dimensional time-dependent systems.…”