Adapted Measures for Markov Interval Maps

TL;DR

This paper investigates the relationship between entropy and adapted invariant measures in Markov interval maps with singularities, establishing conditions under which the measure of maximal entropy is adapted.

Contribution

It introduces a criterion linking entropy and singularity strength to determine when the measure of maximal entropy is adapted in Markov interval maps.

Findings

A specific condition relates entropy and singularity strength to measure adaptation.

Recurrence of the singularity is necessary for nonadapted invariant measures under a H"older condition.

The results extend understanding of measure adaptation in nonuniform hyperbolic systems.

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.