Symbolic extension of ${\mathcal{C}}^{1,\alpha }$ maps in arbitrary dimensions

Abstract

We prove that every ${\mathcal{C}}^{1,\alpha }$ self-map of a compact Riemannian manifold, in arbitrary dimension, admits a symbolic extension. This gives a positive answer to the conjecture of Downarowicz and Newhouse in arbitrary dimensions. Our method is based on analyzing the small singular values of the linearized orbit operator.

arXiv commentThis version reorganizes the historical background of the problem, recalled the definition of "entropy structure", and fixes several typos. 22pages, any comments or suggestions would be highly appreciated