Expanding maps of an interval with holes

Abstract

We study a class of open chaotic dynamical systems. Consider an expanding map of an interval from which a few small open subintervals are removed (thus creating ‘holes’). Almost every point of the original interval then eventually escapes through the holes, so there can be no absolutely continuous invariant measures. We construct a so-called conditionally invariant measure that is equivalent to the Lebesgue measure. Our measure is unique and naturally generates an invariant measure, which is singular. These results generalize early work by Pianigiani, Yorke, Collet, Martinez, and Schmidt, who studied similar maps under an additional Markov assumption. We do not assume any Markov property here and use ‘bounded variation’ techniques rather than Markov coding. Our results supplement those of Keller, who studied analytic interval maps with holes by using different techniques.