Sharp Statistical Properties for a Family of Multidimensional NonMarkovian Nonconformal Intermittent Maps

Abstract

Intermittent maps of Pomeau-Manneville type are well-studied in onedimension, and also in higher dimensions if the map happens to be Markov. In general, the nonconformality of multidimensional intermittent maps represents a challenge that up to now is only partially addressed. We show how to prove sharp polynomial bounds on decay of correlations for a class of multidimensional intermittent maps. In addition we show that the optimal results on statistical limit laws for one-dimensional intermittent maps hold also for the maps considered here. This includes the (functional) central limit theorem and local limit theorem, Berry-Esseen estimates, large deviation estimates, and convergence to stable laws and Lévy processes.

Inducing schemes for multi-dimensional piecewise expanding maps

“…An example of a dynamical system to which this method applies, but previous methods do not, is available in [4]. As shown in [4], the results of the current paper are also relevant to proving statistical properties for non-uniformly expanding dynamical systems.…”

“…An example of a dynamical system to which this method applies, but previous methods do not, is available in [4]. As shown in [4], the results of the current paper are also relevant to proving statistical properties for non-uniformly expanding dynamical systems. This is because through inducing, one can replace the mechanism of non-uniformity of hyperbolicity with the presence of discontinuities.…”

“…This is because through inducing, one can replace the mechanism of non-uniformity of hyperbolicity with the presence of discontinuities. All of this is worked out for a family of a multidimensional, nonMarkov, nonConformal intermittent maps in [4]. Other systems to which the current framework can be applied are "Hu-Vaienti"-type maps [5] and "Viana"-type maps [7,1].…”

Mixing rates for symplectic almost Anosov maps

“…The one dimensional expanding map case, starting with [13] and continuing with [6,18,19], has witnessed many progresses that have developed into a rather satisfactory theory. Recently such a theory has been extended to important multidimensional expanding examples [5,8,9]. In contrast, the study of non-uniformly hyperbolic maps is still unsatisfactory.…”

Mixing rates for symplectic almost Anosov maps

“…The one dimensional expanding map case, starting with [12] and continuing with [16,17,6], has witnessed many progressess that have developed into a rather satisfactory theory. Recently such a theory has been extended to important multidimensional expanding examples [8,9,5]. On the contrary the study of non-uniformly hyperbolic maps is still unsatisfactory.…”