Quasi-invariant measures, escape rates and the effect of the hole

Wael Bahsoun; Christopher Bose

doi:10.3934/dcds.2010.27.1107

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2010, Volume 27, Issue 3: 1107-1121. Doi: 10.3934/dcds.2010.27.1107

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Quasi-invariant measures, escape rates and the effect of the hole

Wael Bahsoun^1, and
Christopher Bose^2,

1.
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU
2.
Department of Mathematics and Statistics, University of Victoria, P.O. Box 3045, Victoria, BC, Canada V8W 3P4

Received: September 30, 2009

Revised: December 31, 2009

Published: August 2010

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Abstract

Let $T$ be a piecewise expanding interval map and $T_H$ be an abstract perturbation of $T$ into an interval map with a hole. Given a number ℓ, 0 < ℓ < l, we compute an upper-bound on the size of a hole needed for the existence of an absolutely continuous conditionally invariant measure (accim) with escape rate not greater than -ln(1-ℓ). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.

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Mathematics Subject Classification: Primary 37A05, 37E05.

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