We prove that C 2 surface diffeomorphisms have symbolic extensions, i.e. topological extensions which are subshifts over a finite alphabet. Following the strategy of T.Downarowicz and A.Maass [15] 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.

Building on the theory of symbolic extensions and uniform generators for discrete transformations we develop a similar theory for topological regular flows. In this context a symbolic extension is given by a suspension flow over a subshift.

For a topological dynamical system (X, T ) we define a uniform generator as a finite measurable partition such that the symmetric cylinder sets in the generated process shrink in diameter uniformly to zero. The problem of existence and optimal cardinality of uniform generators has lead us to new challenges in the theory of symbolic extensions. At the beginning we show that uniform generators can be identified with so-called symbolic extensions with an embedding, i.e., symbolic extensions admitting an equivariant measurable selector from preimages. From here we focus on such extensions and we strive to characterize the collection of the corresponding extension entropy functions on invariant measures. For aperiodic zero-dimensional systems we show that this collection coincides with that of extension entropy functions in arbitrary symbolic extensions, which, by the general theory of symbolic extensions, is known to coincide with the collection of all affine superenvelopes of the entropy structure of the system. In particular, we recover, after [Bu16], that an aperiodic zero-dimensional system is asymptotically hexpansive if and only if it admits an isomorphic symbolic extension. Next we pass to systems with periodic points, and we introduce the notion of a period tail structure, which captures the local growth rate of periodic orbits. Finally, we succeed in precisely identifying the wanted collection of extension entropy functions in symbolic extensions with an embedding: these are all the affine 1. There exists a uniform generator 2 P in (X, T ), of cardinality ;1 See Remark 1.3 for the precise meaning of measurability. 2 Later we will show that the measurability assumption of P can be dropped. That is, the existence of a "non-measurable uniform generator" implies the existence of a measurable one-see Remark 4.15.

We prove that C r maps with r > 1 on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [9].R. -Nous montrons que toute dynamique de classe C r avec r > 1 sur une surface compacte admet une extension symbolique, i.e. une extension topologique qui est un sous-décalage à alphabet fini. Nous donnons plus précisément une borne (optimale) sur l'infimum de l'entropie topologique de toutes les extensions symboliques. Ceci répond positivement à une conjecture de S. Newhouse and T. Downarowicz en dimension deux et améliore un résultat précédent de l'auteur [9].

Link to this article: http://journals.cambridge.org/abstract_S0143385708080425 How to cite this article: DAVID BURGUET (2009). A direct proof of the tail variational principle and its extension to maps. Abstract. Downarowicz [Entropy structure. J. Anal. 96 (2005), 57-116] stated a variational principle for the tail entropy for invertible continuous dynamical systems of a compact metric space. We give here an elementary proof of this variational principle. Furthermore, we extend the result to the non-invertible case.