Devroye inequality for a class of non-uniformly hyperbolic dynamical systems

Abstract

In this paper we prove an inequality which we call the ‘Devroye inequality’ for a large class of non-uniformly hyperbolic dynamical systems (M, f). This class, introduced by Young, includes families of piecewise hyperbolic maps (Lozi-like maps), scattering billiards (e.g. planar Lorentz gas), unimodal and Hénon-like maps. The Devroye inequality provides an upper bound for the variance of observables of the form K(x, f(x), …, fn−1(x)), where K is any separately Hölder continuous function of n variables. In particular, we can deal with observables which are not Birkhoff averages. We will show in Chazottes et al (2005 Nonlinearity 18 2341–64) some applications of Devroye inequality to statistical properties of this class of dynamical systems.