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

We prove that for ${\mathcal{C}}^{1,\alpha }$ diffeomorphisms on a compact manifold $M$ with $\mathrm{d}\mathrm{i}\mathrm{m}M\le 3$, if an invariant measure $\mu$ is a continuity point of the sum of positive Lyapunov exponents, then $\mu$ 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.