Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory, suitable for a variety of one- or two-semester courses. The authors' clear and fluent exposition helps the reader to grasp quickly the most important ideas of the theory, and their use of concrete examples illustrates these ideas and puts the results into perspective. The book requires few prerequisites, with background material supplied in the appendix. The first four chapters cover elementary material suitable for undergraduate students – invariance, recurrence and ergodicity – as well as some of the main examples. The authors then gradually build up to more sophisticated topics, including correlations, equivalent systems, entropy, the variational principle and thermodynamical formalism. The 400 exercises increase in difficulty through the text and test the reader's understanding of the whole theory. Hints and solutions are provided at the end of the book.

Link to this article: http://journals.cambridge.org/abstract_S0143385707001009How to cite this article: KRERLEY OLIVEIRA and MARCELO VIANA (2008). Thermodynamical formalism for robust classes of potentials and non-uniformly hyperbolic maps.Abstract. We develop a Ruelle-Perron-Fröbenius transfer operator approach to the ergodic theory of a large class of non-uniformly expanding transformations on compact manifolds. For Hölder continuous potentials not too far from constant, we prove that the transfer operator has a positive eigenfunction, which is piecewise Hölder continuous, and use this fact to show that there is exactly one equilibrium state. Moreover, the equilibrium state is a non-lacunary Gibbs measure, a non-uniform version of the classical notion of Gibbs measure that we introduce here. Downloaded: 16 Mar 2015 IP address: 128.184.220.23 502 K. Oliveira and M. Vianaproved that equilibrium states can also be obtained as weighted limits of orbital measures supported on periodic orbits. Extension of this approach and of the conclusions beyond the Axiom A context involves some fundamental difficulties, even restricted to non-uniformly hyperbolic systems, that is, systems such that almost every point admits an asymptotically hyperbolic splitting of the tangent space. For one thing, generating Markov partitions are not known to exist in general. Even when they do exist, Markov partitions usually have infinitely many atoms; this leads to considering gases with infinitely many states, a difficult subject not yet well understood.Important contributions have been given recently by several authors: Bruin and Keller [BK98], Denker and Urbański [DU91, DU92, Urb98], Pesin and Senti [PS05], Wang and Young [WY01] for special classes of transformations, such as interval maps, rational functions of the sphere, and Hénon-like maps; Buzzi, Maume and Sarig [Buz99, BMD02, BS03], Sarig [Sar99, Sar01, Sar03], and Yuri [Yur99, Yur00, Yur03], for countable Markov shifts and piecewise expanding maps; and Leplaideur and Rios [LR06] for 'horseshoes with tangencies' at the boundary of hyperbolic systems, to mention just a few of the most recent works. Many of these papers, and particularly [DU92, Sar03, Yur99, Yur00, Yur03], deal with systems having neutral periodic points, a setting of non-hyperbolic dynamics which has attracted a great deal of attention over the last few years. Also, very recently, Buzzi [Buz05] introduced the important notion of entropyexpansiveness, which influenced other works such as [OV06] and [BR06].There has also been substantial progress in the study of physical measures. In particular, Alves et al [ABV00, BV00] proved the existence and uniqueness of SRB measures for some large classes of non-uniformly hyperbolic maps. One important difficulty in this context lies in the very definition of non-uniform hyperbolicity: [ABV00] assume that Lebesgue almost every point has only non-zero Lyapunov exponents, but it is not clear how this kind of condition could be useful when considering more general potentials, since ...

Abstract. We show that, for a robust (C 2 -open) class of random nonuniformly expanding maps, there exists equilibrium states for a large class of potentials.In particular, these sytems have measures of maximal entropy. These results also give a partial answer to a question posed by Liu-Zhao. The proof of the main result uses an extension of techniques in recent works by Alves-Araújo, Alves-Bonatti-Viana and Oliveira.

In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in [12]. Despite the fact that the non-wandering set is a horseshoe, it contains intervals. We prove that every recurrent point has non-zero Lyapunov exponents and all ergodic invariant measures are hyperbolic. As a consequence, we obtain the existence of equilibrium measures for any continuous potential. We also obtain an example of a family of C ∞ potentials with phase transition.

We construct equilibrium states, including measures of maximal entropy, for a large (open) class of non-uniformly expanding maps on compact manifolds. Moreover, we study uniqueness of these equilibrium states, as well as some of their ergodic properties.

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This article documents the addition of 153 microsatellite marker loci to the Molecular Ecology Resources Database. Loci were developed for the following species: Brassica oleracea, Brycon amazonicus, Dimorphandra wilsonii, Eupallasella percnurus, Helleborus foetidus, Ipomoea purpurea, Phrynops geoffroanus, Prochilodus argenteus, Pyura sp., Sylvia atricapilla, Teratosphaeria suttonii, Trialeurodes vaporariorum and Trypanosoma brucei. These loci were cross‐tested on the following species: Dimorphandra coccicinea, Dimorphandra cuprea, Dimorphandra gardneriana, Dimorphandra jorgei, Dimorphandra macrostachya, Dimorphandra mollis, Dimorphandra parviflora and Dimorphandra pennigera.

Abstract. We prove that, under a mild condition on the hyperbolicity of its periodic points, a map g which is topologically conjugated to a hyperbolic map (respectively, an expanding map) is also a hyperbolic map (respectively, an expanding map). In particular, this result gives a partial positive answer for a question done by A. Katok, in a related context.