Entropy Density of Ergodic Nonadapted Measures for Markov Interval Maps

TL;DR

This paper investigates the structure of ergodic measures for Markov interval maps, showing that nonadapted ergodic measures are prevalent and often form a connected set within the space of all ergodic measures.

Contribution

It demonstrates that nonadapted ergodic measures form a residual and often path-connected subset within the space of ergodic measures for Markov interval maps.

Findings

Nonadapted ergodic measures are residual among ergodic measures.

The set of nonadapted ergodic measures is often path connected.

Results are specific to uniformly expanding transitive Markov interval maps.

Abstract

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.