# Parallel Talks

 Name Adriana Da Luz Institution CMAT Title “Star Flow With Singularities Of Different Indices” Abstract A vector field X is called a star flow if every periodic orbit of any vector field C1-close to X is hyperbolic. It is known that an isolated chain recurrent class of a star flow X on a 3 or 4 manifold are either hyperbolic, or singular hyperbolic (Morales, Pacifico, Pujals for 3 manifolds and Ming Li, Shaobo Gan and Lan Wen on 4-manifolds). Moreover, it was recently proven by Yi Shi Shaobo Gan and Lan Wen for every chain recurrent class C of X a star flow X, if all singularities in C have the same index, then the chain recurrent set of X is singular hyperbolic. We present here a non empty open set of star flows on a 5-manifold for which two singular points of different indices belong (robustly) to the same chain recurrence class. This prevent the class to be singular hyperbolic. We also present a weak form of hyperbolicity (called multisingular hyperbolic) which makes compatible the hyperbolic structures of regular orbits together with the one of singularities of index 2 and 3. This is a joint work with Christian Bonatti. Name Katrin Gelfert Institution UFRJ Title “Convex sum of hyperbolic measures” Abstract In the uniformly hyperbolic setting it is well known that the measure supported on periodic orbits is dense in the convex space of all invariant measure. We study the reverse question: assuming that some ergodic measure converges to a convex sum of hyperbolic ergodic measures, what can we deduce on the initial measures? To every hyperbolic measure whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class of periodic orbits for the homoclinic relation, that we call its intersection class. In a dominated setting, we prove that a convex sum of finitely many ergodic hyperbolic measures of the same index is accumulated by ergodic measures if, and only if, they share the same intersection class. We provide examples which indicate the importance of the domination assumption.  This is joint work with Christian Bonatti. Name Rafael Lucena Institution UFRJ Title “Spectral Gap for Contracting Fibers Systems” Abstract We consider transformations preserving a contracting foliation, such that the associated quotient map satisfies a Lasota Yorke inequality. We prove that the associated transfer operator, acting on suitable normed spaces, has spectral gap. The techniques which are used are relatively simple, and need very small regularity assumptions. As an application we prove spectral gap for Lorenz-Like two dimensional maps, having C1+  regularity, possible unbounded expansion and contraction rates. Name Dominique Malicet Institution PUC-Rio Title “Random walks in the group of homeomorphisms of the circle” Abstract In this talk, I will present various results I obtained on the behavior of a general left random walk $g_n=f_{n-1}\cdots f_0$ on the group ofhomeomorphisms/ diffomorphisms of the circle, mainly concerning the assymptotic behavior of  an orbit $(g_n(x))_{n\in\N}$ for a typical sequence $(g_n)$, and the sensivity of this orbit to small perturbations of the initial condition $x$. Name Alexey Okunev Institution Higher School of Economics Title “Milnor attractors of circle skew products” Abstract Partially hyperbolic skew products provide interesting dynamics even for one-dimensional fiber. For the interval fiber there are strange examples in border-preserving case: intermingled basins (I.Kan) and thick (i.e. having positive but not full measure) attractor (Yu.Ilyashenko). In non-border-preserving case there is a finite collection of alternating attractors and repellers, each of them is a so-called bony graph (V.Kleptsyn, D.Volk). The Milnor attractor is defined as the smallest closed subset, containing omega limit sets of almost all points w.r.t. the Lebesgue measure. We will discuss the following result: for a C^2-generic skew product with circle fiber over an Anosov diffeomorphism the Milnor attractor is Lyapunov stable and not thick. Moreover, such skew product is either transitive or has non-wandering set of zero measure. The result is proved under the assumption that the fiber maps preserve the orientation of the circle. Main ingredients of the proof are the semicontinuity lemma and the fact that omega-limit set of a generic point is saturated by the unstable leaves. Name Mario Ponce Institution PUC-Chile Title “The scaling mean and a Law of Large Permanents” Abstract This is a joint work with Jairo Bochi and Godofredo Iommi (PUC-Chile). In this talk we study two types of means of the entries of a non- negative matrix: the permanental mean, which is defined using permanents, and the scaling mean, which is defined in terms of an optimization problem. We explore relations between these two means, making use of important results by Ergorychev and Falikman (the van der Waerden conjecture), Friedland, Sinkhorn, and others. We also define a scaling mean for functions in a much more general context. Our main result is a Law of Large Permanents, a point- wise ergodic theorem for permanental means of dynamically defined matrices that expresses the limit as a functional scaling mean. The concepts introduced in this talk are general enough so to include as particular cases certain classical types of means, as for example symmetric means and Muirhead means. Name Cristóbal Rivas Institution USACH Title “Rigidity of diffeomorphism groups of the interval” Abstract We will review some recent result concerning the degree of differentiability of group action on the closed interval. We will be mainly concerned with nilpotent and solvable groups. Name Yi Shi Institution IMPA Title “Suspended Anosov flows and DA-attractors” Abstract In this talk, we consider the time one map of a suspended Anosov flow of a linear Anosov automorphism. We show that the time one map could be $C^1$-approximated by structurally stable diffeomorphisms which exhibiting a Smale DA-attractor. However, we show that there exists a $C^2$-neighborhood of the time one map does not contain any diffeomorphisms topologically conjugate to them. This implies such kind perturbations for getting DA-attractors could not be $C^2$-small. Name Shirou Wang Institution Peking University Title “Upper Semi-Continuity Of Entropy Map For Nonuniformly Hyperbolic Systems “ Abstract We prove that entropy map is upper semi-continuous for C1 nonuniformly hyperbolic systems with domination, while it is not true for C1+α nonuniformly hyperbolic systems in general. This goes a little against a common intuition that conclusions are parallel between C1+domination systems and C1+α systems. Gang Liao, Wenxiang Sun, Shirou Wang . Name Jiagang Yang Institution U. Federal Fluminense Title “Continuity of entropy” Abstract In this talk, I will explain my recent works on continuation of entropy. References: [1] G.Liao, M.VIANA, J.YANG. The entropy conjecture for diffeomorphisms away from tangencies [2] Y.ZHOU, J.Yang. Generic continuity of metric entropy for volume-preserving diffeomorphisms [3] J.Yang. Topological entropy of Lorenz-like flows [4] R.SAGHIN, J.Yang,  Continuity of topological entropy for perturbations of time-one maps of hyperbolic flows.