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

Bruin, H, Demers, M and Melbourne, I (2007) Existence and convergence properties of physical measures for certain dynamical systems with holes Ergodic Theory and Dynamical Systems, 30 (3). pp. 687-728. (Unpublished)

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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.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
Bruin, H
Demers, M
Melbourne, I
Date : 29 May 2007
DOI : 10.1017/S0143385709000200
Related URLs :
Additional Information : Copyright 2007 Cambridge University Press.
Depositing User : Symplectic Elements
Date Deposited : 03 Jan 2012 15:38
Last Modified : 31 Oct 2017 14:16

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