, Volume 186, Issue 1, pp 191-236
Date: 01 Mar 2011

\(\mathcal{C}^{2}\) surface diffeomorphisms have symbolic extensions

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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 (Invent. Math. 176:617–636, 2009) 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.