Speaker: José Alves (CMUP, Portugal)

Title: Entropy formula for systems with discontinuities

Abstract: We obtain some entropy formulas for the SRB measures of two classes of systems with singular sets. The first class consists ofgeneral endomorphisms displaying nonuniform expansion and slow recurrence to a nondegenerate singular set. The second, inspired on Poincaré return maps of the Lorenz flow, comprises partially hyperbolic diffeomorphisms with singularities which are nonuniformly expanding along the centre-unstable direction and also exhibit slow recurrence to a nondegenerate singular set. Joint work with D. Mesquita.

Speaker: Martin Andersson (UFF, Brazil)

Title: From Bowen's eyes to physical measures in reparameterized linear toral flows with stopping points

Abstract: Consider a constant vector field $X = (1, \alpha)$ with irrational slope on $\mathbb{T}^2$. Multiply $X$ by a smooth function $f$ which is positive everywhere except at two points $p$ and $q$, where it vanishes. Let $\phi^t$ be the flow associated to this vector field. One can show that, for such a flow, the only invariant probability measures are combinations of Dirac measures at the points $p$ and $q$. A natural question arises: what happens to the time averages of $\phi^t(x)$ for a Lebesgue-typical point $x \in \mathbb{T}^2$ as $t$ tends to infinity? In this talk I will give some answers to that question. This is a joint work with Pierre-Antoine Guihéneuf (Université Paris 6, France).

Speaker: Lorenzo J. Díaz (PUC-Rio, Brazil)

Title: Generation of independent homoclinic classes at heterodimensional cycles

Abstract: Newhouse proved that homoclinic tangencies of $C^2$ surface diffeomorphisms generate infinitely many sinks. A sink is an "independent piece of dynamics". This result motivates a similar question for heterodimensional cycles in partially hyperbolic settings. In this setting, there are no sinks and homoclinic classes are "independent pieces of dynamics". We discuss this question and explain the limitations for the creation of "independent homoclinic classes".

Speaker: Cristina Lizana (UFBA, Brazil)

Title: Invariance of Entropy for maps isotopic to Anosov

Abstract: We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of $\mathbb{T}^d$ with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on $H_1(\mathbb{T}^d)$ is hyperbolic. In absence of the simplicity condition we construct a robustly transitive counter-example. This is a joint work with P. Carrasco, E. Pujals and C. Vásquez.

Speaker: Enrique Pujals (CUNY, USA)

Title: TBA

Abstract: TBA

Speaker: Jana Rodríguez-Hertz (SUSTech, China)

Title: Robust minimality of strong foliations for DA diffeomorphisms: new examples

Abstract: Let f be a C^2 partially hyperbolic diffeomorphisms of T^3 (not necessarily volume preserving or transitive) isotopic to a linear Anosov diffeomorphism A with eigenvalues λ_3 < 1< λ_2 < λ_1. If the set { x : | log det( Tf |E^{cu}(f) ) | ≤ log λ_1} has zero volume inside any unstable leaf of f, then the stable foliation of f is C^1 robustly minimal, i.e., the stable foliation of any diffeomorphism C^1 sufficiently close to f is minimal. In particular, f itself is robustly transitive. We build, with this criterion, a new example of a C^1 open set of DA diffeomorphisms, such that the strong stable foliation and the strong unstable foliation of any diffeomorphism in this open set are both minimal. The existence of such an example was unknown in this setting. This is a joint work with R. Ures and J. Yang.

Speaker: Ali Tahzibi (ICMC USP São Carlos, Brazil)

Title: A dichotomy for measures of maximal entropy of discretized Anosov flows

Abstract: Discretized Anosov flows coincide with partially hyperbolic diffeomorphisms homotopic to identity in many 3-dimensional manifolds. In this talk we study measures of maximal entropy for this class and prove some dichotomy results. In particular, we have the following: For any C2 diffeomorphism f, in a C1 neighbourhood of time-one map of the geodesic flow on a negatively curved surface either f admits a unique m.m.e and this measure is non-hyperbolic and f is embedded to a topological flow or there exists exactly 2 ergodic m.m.e which are hyperbolic with opposite central exponent sign. Results are joint work with Buzzi, Crovisier and Poletti.