Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval

Details Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval Markov Extensions for Dynamical Systems with Holes: An Application to Expanding Maps of the Interval

2004-04-13

arxiv.org/abs/math/0404255v3

Israel J. of Math. 146 (2005), 189-221

Mathematics

Dynamical Systems

32 pages. New version contains minor revisions, primarily to clarify introductory Section 2

Scientific paper

We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. for expanding maps of the interval with holes. We obtain the convergence of the a.c.c.i.m. to the SRB measure of the corresponding closed system as the measure of the hole shrinks to zero.

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