On Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps

Abstract

In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our coupling method only requires that the expanding map satisfies three assumptions: (H1) Chernov’s one-step expansion at q-scale; (H2) dynamically Hölder continuity of log Jacobian (on each branch); (H3) eventually covering over a magnet interval, which are much weaker than the standard assumptions for uniformly expanding maps. We further conclude the existence of an absolutely continuous invariant probability measure and establish the regularity of its density function. Moreover, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem) with respect to a large class of unbounded observables that we call “dynamically Hölder series". Our approach is particularly powerful for the piecewise expanding maps which do not satisfy the “big image property" and the maps that have inverse Jacobian of low regularity. As few is known on the statistical properties of such maps in the literature, we demonstrate our results for a specific class of piecewise expanding maps of this kind.