Adapted Measures for Markov Interval Maps

Abstract

Adapted invariant measures, such as the natural area measure (Liouville), have a central place in the development of ergodic theory for billiards. These measures ensure local Pesin charts can be constructed almost everywhere even in the nonuniformly hyperbolic setting. Recently, for Sinai billiards satisfying certain conditions, the unique measure of maximal entropy has been shown to be adapted. However, not all positive entropy measures are. To investigate the connection between entropy and adaptedness, we examine Markov interval maps with exactly one singularity. We prove that a condition relating the entropy of the map and the "strength" of the singularity determines if the measure of maximal entropy is adapted with respect to this singularity. We also show that under a H\"{o}lder condition, recurrence of the singularity is necessary to have nonadapted invariant measures.