Symbolic extensions in intermediate smoothness on surfaces

Abstract

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].

Periodic expansiveness of smooth surface diffeomorphisms and applications

“…The main novelty in the present paper consists in approximating not only the C r map but also the action of its derivative on the unit tangent bundle. A similar approach was developed in [12] to build symbolic extensions of C r smooth surface systems with r > 1. 7.1.…”

Entropy of Physical Measures for $$C^\infty $$ Dynamical Systems

“…Unlike the Main Theorem, which is false in finite smoothness, we conjecture Corollary 4 holds true for any C 1+α map. It can be deduced from the Reparametrization Lemma in [5] the case of C 1+α interval maps and surface diffeomorphisms. However as it involves stronger technicalities we prefer to only consider C ∞ maps in the present paper.…”

“…Moreover the reparametrized set was the whole dynamical ball (here this is the case when f is a local diffeomorphim by choosing a small enough). Others similar forms of the Reparametrization Lemma were used succesfully by the author to study symbolic extensions and exponential growth of periodic points for C r surface diffeomorphisms [5,6]. The technical proof could be skipped at a first reading.…”

“…As in the previous works [24,9,5] we may replace † σ by s = α −1 σ(α·) for α > 0 and the local dynamic of f around x of time n by the nonautonomous…”

“…In particular it allows him to show that Shub's entropy conjecture holds true in this setting. Using a similar approach we manage to control locally not only the volume growth but also the distortion (see also [5] and [6]). The resulting Reparametrization Lemma of dynamical balls is the key argument in the proof of our Main Theorem.…”