Document Type

Article

Article Version

Publisher's PDF

Publication Date

2011

Abstract

We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions, including Lebesgue and Sinai–Reulle–Bowen (SRB) measures. Lower bounds do not hold in such a generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon with holes.

Comments

Copyright 2011 Cambridge University Press

http://journals.cambridge.org/abstract_S0143385711000344

Publication Title

Ergodic Theory and Dynamical Systems

Published Citation

MARK F. DEMERS, PAUL WRIGHT and LAISANG YOUNG (2012, Published online September 2011). Entropy, Lyapunov exponents and escape rates in open systems. Ergodic Theory and Dynamical Systems, 32, pp 1270- 1301 doi:10.1017/S0143385711000344

DOI

10.1017/S0143385711000344

Peer Reviewed

Download

Share

COinS