$C^2$ surface diffeomorphisms have symbolic extensions

Abstract

We prove that (\mathcal{C}^{2}) surface diffeomorphisms have symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. Following the strategy of Downarowicz and Maass we bound the local entropy of ergodic measures in terms of Lyapunov exponents. This is done by reparametrizing Bowen balls by contracting maps in a approach combining hyperbolic theory and Yomdin’s theory.