Dynamical Systems Seminar

Adapted invariant measures, such as the natural area measure (Liouville), have a central place in the development of the ergodic theory for billiards. These measures ensure local Pesin charts may 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, I will discuss Markov interval maps with exactly one singularity. I will show that a condition relating the entropy of the map and the “strength” of the singularity determines if the measure of maximal entropy is adapted. I will also show that under a Hölder condition, recurrence of the singularity is necessary to have nonadapted invariant measures.