### Abstract

We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.

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

Pages (from-to) | 353-388 |

Number of pages | 36 |

Journal | Communications in Mathematical Physics |

Volume | 294 |

Issue number | 2 |

DOIs | |

State | Published - Jan 2010 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*294*(2), 353-388. https://doi.org/10.1007/s00220-009-0941-y

**Escape rates and physically relevant measures for billiards with small holes.** / Demers, Mark; Wright, Paul; Young, Lai-Sang.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 294, no. 2, pp. 353-388. https://doi.org/10.1007/s00220-009-0941-y

}

TY - JOUR

T1 - Escape rates and physically relevant measures for billiards with small holes

AU - Demers, Mark

AU - Wright, Paul

AU - Young, Lai-Sang

PY - 2010/1

Y1 - 2010/1

N2 - We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.

AB - We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.

UR - http://www.scopus.com/inward/record.url?scp=76349119232&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=76349119232&partnerID=8YFLogxK

U2 - 10.1007/s00220-009-0941-y

DO - 10.1007/s00220-009-0941-y

M3 - Article

VL - 294

SP - 353

EP - 388

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -