# Plenary Talks

 Name Pierre Berger Institution Université Paris 13 Title “Generic family with robustly inifitely many sinks” Abstract Given a manifold $M$ of dimension at least 3, $0\le d< r\le \infty$ or $2\le r=d <\infty$,  we show the existence of an open set $U$ of $C^d$-families of $C^r$ diffeomorphisms of $M$ s.t. there exists a Baire residual set $R\subset U$ which satisfies: for every $(f_a)_a \in R$, for every $\|a\|<1$, the diffeomorphism $f_a$ has infinitely many sinks. Name Jairo Bochi Institution PUC-Chile Title “Linear representations and dominated splittings” Abstract A linear representation ρ of a finitely generated group $\Gamma$ into the matrix group $GL(d, \mathbb{R})$ has an exponential gap of index $p$ if the quotient between the p-th and the $p + 1$-th singular values of the matrix $\rho(\gamma)$ grows exponentially with respect to the word-length of $\gamma$. The existence of a representation with such property implies that the group $\Gamma$ is hyperbolic in the sense of Gromov, and moreover that the representation is Anosov in the sense of Labourie, Guichard, and Wienhard. These results were obtained very recently by Kapovich, Leeb, and Porti, answering a question of Gueritaud, Guichard, Kassel, and Wienhard. However, the proof that I shall present, obtained jointly with R. Potrie and A. Sambarino, is much more elementary. We use the notion of dominated splittings for linear cocycles, which comes from differentiable dynamics. An essential tool is a characterization of domination in terms of singular values, obtained by Gourmelon and myself, which generalizes an earlier 2-dimensional result of Yoccoz. Name Christian Bonatti Institution University of Bourgogne Title “Partially hyperbolic diffeomorphisms on 3-manifolds: new examples” Abstract In a sequence of works with Rafael Potrie, Kamlesh Parwani, Andrey Gogolev and Andy Hammerlindl, we build more and more sophisticate examples of partially hyperbolic diffeomorphisms (most of them robustly transitive and stably ergodic) which are not “topologically center leaves equivalent” to one of the classical models, (even “up to finite interation or lift on a finite cover).  This contrdicts classification conjectures by Pujals and by Hertz Hertz  and Ures. I will present these new example, and ask question in the spirit of the classification conjectures. Furthermore, the behaviour of the new examples, beyong their partial hyperbolicity and transitivity/ergodicity are very few understood, leading to numerous natural questions. Name Aaron Brown Institution University of Chicago Title “Entropy, rigidity, and invariant measures for smooth lattice actions” Abstract Consider a smooth action of a lattice in a higher-rank, simple Lie group G on a compact manifold M.  We show that if the dimension of M is sufficiently small relative to the rank of G, then there always exists an invariant probability measure for the action.  If the dimension of M falls in an intermediate range (relative to the rank of G) then we show there exists a quasi-invariant measure such that the action is isomorphic to a relatively measure-preserving extension over a standard boundary action.  The proofs of these results follow from existing measure rigidity techniques combined with a new entropy formula for measures invariant under smooth actions of higher-rank Abelian groups.  This formula establishes a “product structure” (along coarse Lyapunov foliations) of entropy for measures invariant under a smooth action of a higher-rank abelian group.  The product structure of entropy follows, in turn, from a generalization of the Ledrappier-Young entropy formula to “entropy subordinated to a foliation.” Name Clark Butler Institution University of Chicago Title “Continuity and Discontinuity of Lyapunov Exponents for SL(2,R) cocycles with respect to Holder Norms” Abstract We consider the problem of characterizing the continuity points of Lyapunov exponents of $SL(2,\mathbb{R})$-valued cocycles over hyperbolic systems with respect to Holder norms on the space of cocycles. In the $C^0$ norm, a special case of a theorem of Bochi-Mane characterizes the continuity points as the zero exponent cocycles and uniformly hyperbolic cocycles. We show, in sharp contrast to the$C^0$ situation, that in any Holder norm there are open sets of cocycles satisfying a “fiber-bunching” condition which are continuity points for Lyapunov exponents but need not be uniformly hyperbolic and may even be “reducible” in an appropriate generalized sense. This is a joint work with Lucas Backes and Aaron Brown which extends work of Bocker-Viana and Malheiro-Viana on the cases of Bernoulli cocycles and Markov cocycles respectively. We also exhibit a family of cocycles which come arbitrarily close to satisfying the fiber bunching inequality but are discontinuity points for Lyapunov exponents and introduce a new open nonuniform fiber bunching condition which is compatible with both continuity and discontinuity of the Lyapunov exponents. Name Álvaro Daniel Coronel Institution UNAB Title “Sensitive dependence of Gibbs measures at low temperature” Abstract The Gibbs measures of an interaction can behave chaotically as the temperature drops to zero. This was first observed in some XY models in statistical mechanics, and then, in some symbolic systems. In this talk we discuss a related phenomenon that we have observed in some quasi-quadratic families. More precisely, there are parameters exhibiting a sensitive dependence of equilibrium states for the geometric potential, that is, an arbitrarily small perturbation on the parameter can produce significant changes in the low-temperature behavior of its equilibrium states. We show that sensitive dependence is also present in some XY models and in some symbolic systems. Joint work with Juan Rivera-Letelier. Name Sylvain Crovisier Institution Univerité Paris 13 Title “On the density of singular hyperbolic vector fields in dimension three” Abstract We present a dichotomy for vector fields in dimension three: those which are singular hyperbolic or exhibit a homoclinic tangency form a dense subset of the space of C1-vector fields. This answers a conjecture by Palis. The argument uses an extension for local fibred flows of Mañé and Pujals–Sambarino’s theorems about the uniform contraction of one-dimensional dominated bundles. This is a joint work with Dawei Yang. Name Todd Fisher Institution Brigham Young University Title “Unique equilibrium states for geodesic flows in nonpositive curvature” Abstract The geodesic flow for a compact Riemannian manifold with negative curvature has a unique equilibrium state for every Holder continuous potential function. This is no longer true if the curvature is only nonpositive. We show that there is a large class of potentials with unique equilibrium states. Specifically, we prove that for compact rank 1 surfaces of nonpositive curvature that the a scalar times geometric potential has a unique equilibrium state for the scalar less than 1. Furthermore, if a potential satisfies a bounded range hypothesis for compact rank 1 manifolds with nonpositive curvature, then there will be a unique equilibrium state. This is joint work with Keith Burns, Vaughn Climenhaga, and Dan Thompson. Name John Franks Institution Northwestern University Title “Some results on entropy zero surface diffeomorphisms” Abstract We consider properties of smooth area preserving diffeomorphisms of compact surfaces (primarily $S^2$) and groups of such diffeomorphisms.  In particular we are interested in the topological structure of invariant sets for such diffeomorphisms and the algebraic properties of the restriction of group to some invariant set (e.g. a finite invariant set).  This represents some joint work with Michael Handel and other joint work with Kamlesh Parwani. Name Francoise Ledrappier Institution University of Notre Dame Title “Rayleigh quotients and equivariant family of measures” Abstract We describe an equivariant family of measures on the boundary at in nity of the universal cover of a compact negatively curved manifold. This family is naturally associated with the bottom of the spectrum of the Laplacian. Name Carlos Gustavo Moreira Institution IMPA Title “On the fractal geometry of horseshoes in arbitrary dimensions” Abstract We will discuss some works in progress which give a reasonably good perspective of understanding the main properties of the fractal geometry of typical dissipative horseshoes in arbitrary dimensions. In the first work, in collaboration with J. Palis and M. Viana, given a horseshoe Λ whose stable spaces have dimension k we define a family of fractal dimensions (the socalled upper stable dimensions) d (j)s (Λ), 1 ≤ j ≤ k which satisfy d (1)s (Λ) ≥ d(2)s(Λ) ≥ · · · ≥d(k) s (Λ) ≥ HD(Λ ∩ Ws (x)), ∀x ∈ Λ (and analogously for the unstable directions) with the following propertie: given 1 ≤ r ≤ k and ε > 0 there is a ε−small C∞ perturbation of the original diffeomorphism for which the hyperbolic continuation of Λ has a subhorseshoe Λ which has strong stable foliations of codimensions ˜ j for 1 ≤ j ≤ r and which satisfies d(r)s(Λ) ˜ > d(r)s(Λ) − ε. In the second work in progress, in collaboration with W. Silva (which extends a previous joint work in codimension 1), we prove that if a horseshoe Λ has strong stable foliations of codimensions j for 1 ≤ j ≤ r and satisfies d(r)s(Λ) > r then it has a small C∞ perturbation which contains a blender of codimension k: in particular C1 images of stable Cantor sets of it (of the type Λ∩Ws (x)) in R k will typically have persistently non-empty interior. We also expect to prove that when r < d (r) s (Λ) ≤ r+ 1 the Hausdorff dimension of these stable Cantor sets typically coincide with d (r+1)s (Λ), and this dimension depends continuously on Λ on these assumptions, which would imply typical continuity of Hausdorff dimensions of stable and unstable Cantor sets of horseshoes. Name José Vieitez Institution Universidad de la República Title “On the concept of expansiveness and generalizations” Abstract We compare different generalizations of the well known concept of expansiveness. We prove that for M a closed surface, f : M → M 2-expansive and Ω(f) = M implies f is 1-expansive. We give examples showing that Ω(f) = M cannot be dropped. As Artigue has shown these examples can be done C2 robust. As a consequence for r ≥ 2 there are Cr -robustly cw-expansive surface diffeomorphisms that are not Anosov. On the other hand on C1-topology Sakai (1997) showed that C1robust cw-expansive on surfaces are Anosov.