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

  • HENK BRUIN (a1), MARK DEMERS (a2) and IAN MELBOURNE (a1)
Abstract
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.

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[2]V. Baladi . Positive Transfer Operators and Decay of Correlations (Advanced Series in Nonlinear Dynamics, 16). World Scientific, Singapore, 2000.

[4]R. Bowen . Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms (Lecture Notes in Mathematics, 470). Springer, Berlin, 1975.

[10]N. N. Cencova . Statistical properties of smooth Smale horseshoes. Mathematical Problems of Statistical Mechanics and Dynamics. Ed. R. L. Dobrushin . Reidel, Dordrecht, 1986, pp. 199256.

[16]P. Collet , S. Martinez and B. Schmitt . Quasi-stationary distribution and Gibbs measure of expanding systems. Instabilities and Nonequilibrium Structures V. Eds. E. Tirapegui and W. Zeller . Kluwer, Dordrecht, 1996, pp. 205219.

[28]W. de Melo and S. van Strien . One Dimensional Dynamics (Ergebnisse Series, 25). Springer, Berlin, 1993.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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