# American Institute of Mathematical Sciences

August  2010, 27(3): 1107-1121. doi: 10.3934/dcds.2010.27.1107

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

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

Received  October 2009 Revised  January 2010 Published  March 2010

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 , 0 < < 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-). The two main ingredients of our approach are Ulam's method and an abstract perturbation result of Keller and Liverani.
Citation: Wael Bahsoun, Christopher Bose. Quasi-invariant measures, escape rates and the effect of the hole. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1107-1121. doi: 10.3934/dcds.2010.27.1107
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