Upper semi-continuity of metric entropy for $\mathcal{C}^{1,α}$ diffeomorphisms
Chiyi Luo, Dawei Yang
TL;DR
The paper proves that for $C^{1,\alpha}$ diffeomorphisms on compact manifolds with $\dim M\le 3$, if an invariant measure $\mu$ is a continuity point of the sum of positive Lyapunov exponents $\lambda_\Sigma^+(\mu,f)$, then $\mu$ is an upper semicontinuity point of the entropy map $h_\mu(f)$. The core method combines Burguet's reparametrization lemma with a finely tuned entropy bound for ergodic measures with a single positive Lyapunov exponent, enabling control of local entropy via a finite partition and a bounded unstable curve. By carefully decomposing measures and exploiting continuity properties of Lyapunov data, the authors deduce USC of entropy in dimension three and on surfaces, with notable corollaries on the dimensions of measures and SPR-type properties for surface diffeomorphisms. These results advance the understanding of how Lyapunov stability influences entropy and related dimension spectra, and they provide a framework for analyzing maximal entropy measures in low dimensions.
Abstract
We prove that for $\mathcal{C}^{1,α}$ diffeomorphisms on a compact manifold $M$ with ${\rm dim} M\leq 3$, if an invariant measure $μ$ is a continuity point of the sum of positive Lyapunov exponents, then $μ$ is an upper semi-continuity point of the entropy map. This gives several consequences, such as the upper-semi continuity of dimensions of measures for surface diffeomorphisms. Furthermore, we know the continuity of dimensions for measures of maximal entropy.