Ergodic Theory and Dynamical Systems

Research Article

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

HENK BRUINa1, MARK DEMERSa2 and IAN MELBOURNEa1

a1 Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK (email: h.bruin@surrey.ac.uk, ism@math.uh.edu)

a2 Department of Mathematics and Computer Science, Fairfield University, Fairfield, CT 06824, USA (email: mdemers@fairfield.edu)

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 μ (a.c.c.i.m.) with the physical property that strictly positive Hölder continuous functions converge to the density of μ under the renormalized dynamics of the system. In addition, we construct an invariant measure ν, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for Hölder observables. We show that ν 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 a.c.c.i.m. 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.

(Received September 15 2008)

(Revised February 17 2009)