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Existence and convergence properties of physical measures for certain dynamical systems with holes
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Bruin, Henk, Demers, Mark and Melbourne, Ian (2010) Existence and convergence properties of physical measures for certain dynamical systems with holes. Ergodic Theory and Dynamical Systems, Vol.30 (No.3). pp. 687-728. doi:10.1017/S0143385709000200 ISSN 0143-3857.
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Official URL: http://dx.doi.org/10.1017/S0143385709000200
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.
| Item Type: | Journal Article | ||||
|---|---|---|---|---|---|
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
| Journal or Publication Title: | Ergodic Theory and Dynamical Systems | ||||
| Publisher: | Cambridge University Press | ||||
| ISSN: | 0143-3857 | ||||
| Official Date: | 2010 | ||||
| Dates: |
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| Volume: | Vol.30 | ||||
| Number: | No.3 | ||||
| Page Range: | pp. 687-728 | ||||
| DOI: | 10.1017/S0143385709000200 | ||||
| Status: | Peer Reviewed | ||||
| Publication Status: | Published | ||||
| Access rights to Published version: | Restricted or Subscription Access |
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