### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 1270-1301 |

Number of pages | 32 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 32 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2012 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

*32*(4), 1270-1301. https://doi.org/10.1017/S0143385711000344

**Entropy, Lyapunov exponents and escape rates in open systems.** / Demers, Mark F.; Wright, Paul; Young, Lai-Sang.

Research output: Contribution to journal › Article

*Ergodic Theory and Dynamical Systems*, vol. 32, no. 4, pp. 1270-1301. https://doi.org/10.1017/S0143385711000344

}

TY - JOUR

T1 - Entropy, Lyapunov exponents and escape rates in open systems

AU - Demers, Mark F.

AU - Wright, Paul

AU - Young, Lai-Sang

PY - 2012/8

Y1 - 2012/8

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84862727121&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84862727121&partnerID=8YFLogxK

U2 - 10.1017/S0143385711000344

DO - 10.1017/S0143385711000344

M3 - Article

AN - SCOPUS:84862727121

VL - 32

SP - 1270

EP - 1301

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 4

ER -