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arXiv · 2504.07746

Upper semi-continuity of metric entropy for $\mathcal{C}^{1,\alpha}$ diffeomorphisms

Chiyi Luo·Dawei Yang
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Abstract

We establish a uniform approximation of metric entropy by partition entropy using uniform partitions for $\cC^{1,\alpha}$ three-dimensional diffeomorphisms. This gives several consequences for diffeomorphisms on a compact manifold $M$ with ${\rm dim} M\leq 3$. First, if an invariant measure $\mu$ is a continuity point of the sum of its positive Lyapunov exponents, then $\mu$ is an upper semi-continuity point of the entropy map. Second, it provides a slight improvement of the necessary condition for strong positive recurrence of surface and three-dimensional diffeomorphisms. Third, it yields continuity of dimensions for measures of maximal entropy.

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Chiyi Luo, Dawei Yang. 2025-04-10. Upper semi-continuity of metric entropy for $\mathcal{C}^{1,\alpha}$ diffeomorphisms. https://arxiv.org/abs/2504.07746

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