Upper semi-continuity of metric entropy for diffeomorphisms with dominated splitting

Abstract: For a $C^{r}$ $(r>1)$ diffeomorphism on a compact manifold that admits a dominated splitting, this paper establishes the upper semi-continuity of the entropy map. More precisely, this paper establishes the upper semi-continuity of the entropy map in the following two cases: (1) if a sequence of invariant measures has only positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the upper limit of their metric entropies is less than or equal to the entropy of the limiting measure; (2) if an invariant measure has positive Lyapunov exponents along a sub-bundle and non-positive Lyapunov exponents along another sub-bundle, then the entropy map is upper semi-continuous at this measure.