Markov extensions for dynamical systems with holes: an application to expanding maps of the intervalIsrael Journal of Mathematics
AbstractWe 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. forC 1+α 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.
Published CitationMark Demers, "Markov extensions for dynamical systems with holes: an application to expanding maps of the interval," Israel Journal of Mathematics 146:1 (2005), 189-221.
Citation InformationMark Demers. "Markov extensions for dynamical systems with holes: an application to expanding maps of the interval" Israel Journal of Mathematics Vol. 146 Iss. 1 (2005)
Available at: http://works.bepress.com/mark_demers/9/