Existence and convergence properties of physical measures for certain dynamical systems with holes

TL;DR

This paper investigates the existence and properties of physical and invariant measures in certain dynamical systems with holes, including expanding and Collet-Eckmann maps, revealing new insights into escape rates and measure convergence.

Contribution

It introduces a framework for analyzing measures in non-uniformly hyperbolic systems with holes, extending thermodynamic formalism to these complex cases.

Findings

Existence of an absolutely continuous conditionally invariant measure with physical properties.

Construction of an ergodic invariant measure with exponential decay of correlations.

Validation of the escape rate formula beyond uniformly hyperbolic systems.

Abstract

We study two classes of dynamical systems with holes: expanding maps of the interval and Collet-Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure $\mu$ (a.c.c.i.m.) with the physical property that strictly positive H\"{o}lder continuous functions converge to the density of $\mu$ under the renormalized dynamics of the system. In addition, we construct an invariant measure $\nu$, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for H\"{o}lder observables. We show that $\nu$ satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet-Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the \accim and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes.