The Upper Semi-continuity Of The Metric Entropy Of Systems With Dominated Splitting

Let f be a C¹ diffeomorphism on a compact Reimannian manifold M, and Λ be a compact invariant subset. If f admits a dominated splitting TΛM=E^cu⊕>E¹⊕>E²···⊕>E^l⊕>E^cs with dim Eⁱ= 1（1 ≤ i ≤ l） and satisfies.Then f is asymptotically entropy expansive, and the topological entropy is upper semi-continuous（w.r.t systems）.