arXiv subjects
Publications and source records attributed to Łukasz Krzywoń.
Given a uniformly expanding transitive Markov interval map, we show that within the set of ergodic measures the set of nonadapted ergodic measures is residual in with respect to the topology induced by the $\overline{d}$-metric. This set of measures is also shown to be path connected in many cases.
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 Hölder condition, recurrence of the singularity is necessary to have nonadapted invariant measures.