Measures of maximal entropy on subsystems of topological suspension semiflows
Volume 260 / 2021
Given a compact topological dynamical system $(X, f)$ with positive entropy and upper semicontinuous 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 semiflow 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.