Measures of maximal entropy on subsystems of topological suspension semi-flows

TL;DR

This paper characterizes measures of maximal entropy for suspension semi-flows over topological dynamical systems, showing they correspond to lifts of measures maximizing entropy on invariant subsets, with implications for the size of these sets.

Contribution

It constructs specific roof functions to precisely describe the measures of maximal entropy for suspension flows, revealing their possible cardinalities.

Findings

Measures of maximal entropy are lifts of entropy-maximizing measures on invariant subsets.

The set of ergodic measures of maximal entropy can be countable, uncountable, or finite.

The results apply to suspension flows over full shifts with various entropy properties.

Abstract

Given a compact topological dynamical system (X, f) with positive entropy and upper semi-continuous entropy map, and any closed invariant subset $Y \subset X$ with positive entropy, we show that there exists a continuous roof function such that the set of measures of maximal entropy for the suspension semi-flow over (X,f) consists precisely of the lifts of measures which maximize entropy on Y. This result has a number of implications for the possible size of the set of measures of maximal entropy for topological suspension flows. In particular, for a suspension flow on the full shift on a finite alphabet, the set of ergodic measures of maximal entropy may be countable, uncountable, or have any finite cardinality.