Let $(X,d)$ be a compact [metric space](/page/Metric%20Space) and let $T:X\to X$ be an expansive homeomorphism with finite topological entropy. Then the entropy map $\mu\mapsto h_\mu(T)$ is upper semicontinuous on $\mathcal M_T(X)$.
Knowledge Status
Analysis
Discussion
States and proves Expansive Systems Have Upper Semicontinuous Entropy Map, a result in advanced ergodic theory focused on entropy, dynamical structure, and related invariants.