We prove that for $\mathcal{ C}^{ 1,\alpha}$ diffeomorphisms on a compact manifold $M$ with ${ \rm dim} M\leq 3$, 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.
Categories: math.DS