Approximation Property On Entropies For Surface Diffeomorphisms

We prove that for any C1 surface diffeomorphism f on a compact 2-dimensional C∞ Riemannian manifold M with positive topological entropy,there exists a diffeomor-phism g arbitrarily close(in the C1 topology)to f exhibiting a horseshoe Λ,such that the topological entropy of g restricted on Λ can arbitrarily approximate the topological entropy of f.This extends a classical result[8,Corollary 4.3]of Katok for C1+α(α>0)surface diffeomorphisms.For the proof,we use the following strategy.If the system admits a dominated splitting on some subset,we can directly use a result of Gelfert[7,Theorem 1]to show the relation between the entropy of the system and the entropy on the horseshoes.Else if the system does not admit an ideal dominated splitting,we then use the theory of Buzzi,Crovisier and Fisher[4,Theorem 4.1]to find a horseshoe with good properties which will eventually lead us prove our main result.