Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems

Details Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems Devroye Inequality for a Class of Non-Uniformly Hyperbolic Dynamical Systems

2004-12-08

arxiv.org/abs/math/0412166v2

Mathematics

Dynamical Systems

Corrected version; To appear in Nonlinearity

Scientific paper

10.1088/0951-7715/18/5/023

In this paper, we prove an inequality, which we call "Devroye inequality", for a large class of non-uniformly hyperbolic dynamical systems (M,f). This class, introduced by L.-S. Young, includes families of piece-wise hyperbolic maps (Lozi-like maps), scattering billiards (e.g., planar Lorentz gas), unimodal and H{\'e}non-like maps. Devroye inequality provides an upper bound for the variance of observables of the form K(x,f(x),...,f^{n-1}(x)), where K is any separately Holder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in \cite{CCS} some applications of Devroye inequality to statistical properties of this class of dynamical systems.

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