Approximation property on entropies for surface diffeomorphisms

Abstract

In this paper, we prove that for any C 1 C^1 surface diffeomorphism f f with positive topological entropy, there exists a diffeomorphism g g arbitrarily close (in the C 1 C^1 topology) to f f exhibiting a horseshoe Λ \Lambda , such that the topological entropy of g g restricted on Λ \Lambda can arbitrarily approximate the topological entropy of f f . This extends a classical result of Katok for C 1 + α ( α > 0 ) C^{1+\alpha }(\alpha >0) surface diffeomorphisms.