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

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, 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 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.
Keywords: Transfer Operator, Interval Maps, Ulam's Method., Escape Rates.
Mathematics Subject Classification: Primary 37A05, 37E0.
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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