- Carlos Cabrera:
*On Poincaré extensions of rational maps*. - Gabriel Calsamiglia:
*On the space of branched projective structures with Fuchsian holonomy*. - Mauricio Correa:
*On The Singular Scheme of split foliations*. - Pablo Dávalos:
*On the sublinear displacement for torus homeomorphisms*. - Samuel Estala:
*Distribution of cusp sections in the Hilbert modular orbifold*. - Thiago Fassarella:
*On the polar degree of projective hypersurfaces*. - Ricardo Gómez Aíza:
*Clasificación de sistemas dinámicos simbólicos*. - Leonardo Macarini:
*Dynamical convexity and elliptic orbits for Reeb flows*. - Roberto Markarian:
*Coupling in billiards with small random perturbations*. - Andrés Navas:
*Nilpotencia y crecimiento para grupos de difeomorfismos*. - María José Pacífico:
*Beyond Expansiveness*. - Javier Ribon:
*Real analytic classification of local holomorphic diffeomorphisms*. - Rafael Ruggiero:
*The Hopf conjecture for k-basic Finsler two-tori without conjugate points*. - Martín Sambarino:
*Dynamical coherence for partially hyperbolic diffeomorphisms istopic to Anosov*. - David P. Sanders:
*Dynamical Systems from the point of view of Experimental Mathematics*. - Marcio Soares:
*Hypersurfaces invariant by Pfaff systems*. - Pablo Suárez Serrato:
*Entropía y flujos geométricos*. - Alí Tahzibi:
*Minimal yet measurable foliations*. - Fabio Armando Tal:
*Conservative homeomorphisms of the 2-torus and related chaotic and elliptic regions*. - Carlos H. Vásquez:
*Partially hyperbolic diffeomorphisms whose central direction is mostly expanding*. - Alberto Verjovsky:
*Hedlund's theorem for compact laminations by hyperbolic Riemann surfaces*. - Jorge Vitorio Pereira:
*Effective Liouvilian Integration*. - Juliana Xavier:
*Actions of solvable Baumslag-Solitar groups on higher genus surfaces*.

**Carlos Cabrera**

*On Poincaré extensions of rational maps*

There is a classical extension, of Mobius automorphisms of the Riemann sphere into isometries of the hyperbolic space \(\mathbb{H}^3\), which is called the Poincaré extension. In this talk, we construct extensions of rational maps on the Riemann sphere over endomorphisms of \(\mathbb{H}^3\) exploiting the fact that any holomorphic covering between Riemann surfaces is Mobius for a suitable choice of coordinates.

We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps.

This is a joint work with Peter Makienko and Guillermo Sienra.

**Gabriel Calsamiglia**

*On the space of branched projective structures with Fuchsian holonomy*

Branched projective structures over surfaces arise naturally in a variety of contexts that we will present and motivate the study of the space of all such structures with some fixed data. As a first result we will show that the space of branched projective structures with fixed Fuchsian holonomy and even positive branching order over a compact orientable surface of genus bigger than two is connected. The result is in deep contrast with the case of branching order zero, where the space is infinite discrete, as was shown by Goldman. In both cases the proof is constructive and uses different types of surgeries to produce every possible structure under consideration.

The result is part of a a joint work with B. Deroin and S. Francaviglia.

**Mauricio Correa**

*On The Singular Scheme of split foliations*

We prove that the tangent sheaf of a codimension one locally free distribution splits as a sum of line bundles if and only if its singular scheme is arithmetically Cohen-Macaulay. In addition, we show that a foliation by curves is given by an intersection of generically transversal holomorphic distributions of codimension one if and only if its singular scheme is arithmetically Buchsbaum. Finally, we establish that these foliations are determined by their singular schemes, and deduce that the Hilbert scheme of certain arithmetically Buchsbaum schemes of codimension 2 is birational to a Grassmannian.

This is a joint work with Marcos Jardim and Renato Vidal.

**Pablo Dávalos**

*On the sublinear displacement for torus homeomorphisms*

For a torus homeomoprphism \(f:\mathbf T^2 \rightarrow \mathbf T^2\) homotopic to the identity, one may define a topological invariant \(\rho(f)\subset {H}_1(\mathbf T^2,\mathbf R)\approx \mathbf R^2\), called the *rotation set*. It is defined via some lift \(\tilde{f}:\mathbf R^2\rightarrow\mathbf R^2\) as the 'assymptotic means of displacement', given by the accumulation points of sequences of the form $$ \left( \frac{ \tilde{f}^{n_i}(x_i) - x_i }{ n_i } \right)_{i\in\mathbf N}$$ where \(n_i\rightarrow\infty\) and \(x_i\in\mathbf R^2\).

It turns out that the rotation set does not capture \(\textit{all}\) the rotation information: there are examples of irrotational maps (rotation set \(= \{(0,0)\}\)) with unbounded orbits in every direction (for the lift \(\tilde{f}\)).

In this talk we will see that this 'sublinear' behavior does not exist for some larger rotation sets, like a compact segment with rational slope containing rationals, or a polygon with rational endpoints, and therefore in this cases the rotation set does capture all the rotation information.

**Samuel Estala**

*Distribution of cusp sections in the Hilbert modular orbifold*

Let \(K\) be a number field, let \(M\) be the Hilbert modular orbifold of \(K\), and let \(m_q\) be the probability measure uniformly supported on the cusp cross sections of \(M\) at height \(q\). When \(K\) is the rational numbers, \(M\) is the clasical modular orbifold and D. Zagier has shown that the distribution of \(m_q\) when \(q\) tends to zero is related to the Riemann hypothesis. I will describe Zagier's method in the case of a general Hilbert modular orbifod \(M\). In this case \(m_q\) distributes uniformly with respect to the normalized Haar measure \(m\) on M as q tends to zero, and also there is a relation between the rate by which \(m_q\) approaches \(m\) and the Riemann hypothesis for the Dedekind zeta function of \(K\).

**Thiago Fassarella**

*On the polar degree of projective hypersurfaces*

Given a hypersurface in the complex projective space we prove several known formulas for the degree of its polar map by purely algebro-geometric methods. Furthermore, we give formulas for the degree of its polar map in terms of the degrees of the polar maps of its components. As an application, we classify the plane curves with polar map of low degree, including a very simple proof of I. Dolgachev’s classification of homaloidal plane curves.

Joint work with Nivaldo Medeiros.

**Ricardo Gómez Aíza**

*Clasificación de sistemas dinámicos simbólicos*

Daremos un panorama de resultados y problemas de clasificación de sistemas dinámicos simbólicos, desde conjugaciones hasta isomorfismos que satisfacen condiciones finitarias.

**Leonardo Macarini**

*Dynamical convexity and elliptic orbits for Reeb flows*

We give a notion of dynamical convexity for general contact manifolds that generalizes the definition for the tight sphere introduced by Hofer-Wysocki-Zehnder. Let \(M\) be a contact manifold given by the prequantization of a closed symplectic manifold. It carries an obvious \({\mathbb Z}_2\)-action which, in the case of the sphere, is generated by the antipodal map. Under some extra assumptions, we prove that the Reeb flow of a dynamically convex \({\mathbb Z}_2\)-invariant contact form on \(M\) has an elliptic closed orbit. This generalizes a result due to Dell'Antonio-D'Onofrio-Ekeland on the existence of elliptic orbits on symmetric convex hypersurfaces in \({\mathbb R}^{2n}\).

This is joint work with Miguel Abreu.

**Roberto Markarian**

*Coupling in billiards with small random perturbations*

A first attempt to obtain results on *coupling* for stochastic billiards on convex tables, with small perturbations on the angles.

Work in progress with Leonardo Rolla (IMPA),

Vladas Sidoravicius (IMPA), María Eulalia Vares (UFRJ)

**Andrés Navas**

*Nilpotencia y crecimiento para grupos de difeomorfismos*

Nos concentraremos en torno a resultados finos que relacionan ya sea el crecimiento o el grado de nilpotencia de un grupo con la regularidad máxima con la que puede actuar sobre el intervalo. Para esto, relacionaremos ideas provenientes de la dinámica unidimensional, las caminatas aleatorias sobre grupos, y el análisis sobre espacios métricos discretos.

**María José Pacífico**

*Beyond Expansiveness*

We show that diffeomorphisms in a residual subset far from homoclinic tangencies are measure expansive. We also show that surface diffeomorphisms presenting homoclinic tangencies can be \(C^1\)-approximated by non-measure expansive diffeomorphisms.

**Javier Ribon**

*Real analytic classification of local holomorphic diffeomorphisms*

We are interested in the dynamics of local complex analytic diffeomorphisms in dimension \(1\). We study the classification modulo conjugacy by germs of homeomorphism that are real analytic in a pointed neighborhood of the origin. Such maps appear naturally in the clasification of unfoldings. More precisely, a map conjugating non-topologically trivial unfoldings \(f_{x}(z)\) and \(g_{x}(z)\) of tangent to the identity maps \(f_{0}(z)\) and \(g_{0}(z)\) has a restriction to the special parameter \(x=0\) that is real analytic outside of the origin [Ribon, arxiv.org/abs/1206.2886, 2012]. Moreover, the real analytic classification is not very interesting.

It is well-known that two local holomorphic diffeomorphisms are conjugated by a real analytic diffeomorphism if and only if they are holomorphically or anti-holomorphically conjugated (cf. [Rey, Thesis, U. Paul Sabatier, 1996] for the tangent to the identity case).

A natural question is wether or not the new classification coincides with the classification modulo conjugacy by holomorphic and anti-holomorphic diffeomorphisms. This is the case for non-resonant diffeomorphisms with indifferent fixed points. The proof is of topological nature and it does not depend on diophantine properties of the linear part. Nevertheless the situation is completely different in the resonant case. The conjugacy classes are not the classes of the holomorphic classification. We will explain a method to build non-trivial examples of local diffeomorphisms conjugated by a diffeomorphism that is real analytic off the origin.

**Rafael Ruggiero**

*The Hopf conjecture for k-basic Finsler two-tori without conjugate points*

We show that analytic, k-basic Finsler two tori without conjugate points have zero flag curvature. This is a Finsler version of the well known Hopf conjecture proved by Hopf for two dimensional Riemannian tori and by Burago-Ivanov for Riemannian tori of any dimension. The k-basic assumption, namely, the flag curvature does not depend on vertical variables, is crucial since Hopf conjecture is known to be false in the Finsler category (Busemann and many others examples...) The proof is a combination of two results. First of all in a joint work with J. Barbosa Gomes we show that k-basic FInsler metrics on two tori which are \(C^{1.L}\) integrable are flat. Then, in a joint work with J. Barbosa Gomes and Mario J. DIas Carneiro we show that analytic, k-basic Finsler metrics on two tori without conjugate points are analytically integrable.

**Martín Sambarino**

*Dynamical coherence for partially hyperbolic diffeomorphisms isotopic to Anosov*

We prove that a partially hyperbolic diffeomorphism which is isotopic (through a isotopy by partially hyperbolic diffeomorphisms) to a linear Anosov diffeomorphism on the d-dimensional torus is dynamicallly coherent. We apply the above to study measures of maximal entropy in this setting.

This is a joint work with T. Fisher and R, Potrie.

**David P. Sanders**

*Dynamical Systems from the point of view of Experimental Mathematics*

I will present a few examples of topics from Dynamical Systems from the point of view of *Experimental Mathematics*, i.e. using computations to explore mathematics and suggest new theorems.

Examples I will discuss include: visualizing the Arnold cat map; finding periodic orbits in the standard map; calculating Lyapunov exponents of nonelastic billiard models.

The examples will be presented using the IPython Notebook. This is a tool for recording an entire computational process, including text, equations, figures and videos, in one single executable document. The IPython Notebook represents a paradigm shift in the workflow of Experimental Mathematics.

**M. G. Soares**

*Hypersurfaces invariant by Pfaff systems*

We consider the question of algebraic integrability of pfaff systems on complex manifolds. Improvements of some known results on this question are presented. This is joint work with M. Correa Jr.

**Pablo Suárez Serrato**

*Entropía y flujos geométricos*

Veremos una reseña de los resultados existentes del comportamiento de la entropía topológica del flujo geodésico a lo largo de algunos flujos geométricos en el espacio de métricas Riemannianas.

Para el caso de superficies de área infinita cuyas geodésicas cerradas permanecen dentro de un compacto (llamadas convexas cocompactas, por D. Sullivan) veremos como éstas técnicas de flujos geométricos nos permiten demostrar propiedades de rigidez dentro de clases conformes de métricas. Obteniendo un analogo de un resultado ya clásico de Katok para superficies compactas, en el que las métricas hiperbólicas se exhiben como minimizantes para la entropía.

Trabajo en colaboración con S. Tapie, Universidad de Nantes.

**Ali Tahzibi**

*Minimal yet measurable foliations*

In this talk we study the disintegration of Lebesgue measure along Central foliation of partially hyperbolic diffeomorphisms On 3-torus. We introduce an Open set of such diffeomorphisms which admit minimal yet measurable foliations . By a measurable foliation we mean that the corresponding Sigma-algebra is countably generated in the sense of volume measure of 3_torus.

**Fabio Armando Tal**

*Conservative homeomorphisms of the 2-torus and related chaotic and elliptic regions*

We will talk about nonwandering homeomorphisms of the two-dimensional torus which are homotopic to the identity and strictly toral, in the sense that they exhibit dynamical properties that are not present in homeomorphisms of the annulus or the plane. This includes all homeomorphisms which have a rotation set with nonempty interior, and is a generic condition for area-preserving homeomorphisms. We present, for these homeomorphisms, a topological partition of the torus into an essential chaotic region and periodic bounded topological disks. Furthermore, we show that this chaotic region is externally transitive, has an abundance of periodic points and the local rate of linear diffusion in the lift is everywhere the same.

We will also discuss some consequences of these results and generalizations to surfaces of higher genus.

Joint work with A. Koropecki (UFF-Brazil)

**Carlos H. Vásquez**

*Partially hyperbolic diffeomorphisms whose central direction is mostly expanding*

Typically, the existence of physical measures for partially hyperbolic diffeomorphisms depends of a condition of abundance of non-zero Lyapunov exponents along the central direction. In the cases studied by Dolgopyat, Bonatti and Viana, they assume abundance of negative Lyapunov exponents along the center direction (mostly contracting condition). Mostly contracting condition was later shown to be \(C^2\) robust, with most of its members satisfying a strong kind of statistical stability.

Mostly expanding condition (like dual of "mostly contracting" condition) seemed not to be enough to ensure the existence of physical measures. That is the reason because Alves, Bonatti and Viana used the condition of non-uniform hyperbolicity along the center direction to deal with the case of positive center Lyapunov exponents.

In this joint work with Martin Anderson, we prove that among the partially hyperbolic diffeomorphisms (in the narrow sense), the mostly expanding condition in fact guarantee the existence of physical measures and provide more information about the statistic of the systems.

**Alberto Verjovsky**

*Hedlund's theorem for compact laminations by hyperbolic Riemann surfaces*

If L is a compact minimal lamination by surfaces of negative curvature, we give a sufficient condition for the horocycle flow on its unit tangent bundle to be minimal.

**Jorge Vitorio Pereira**

*Effective Liouvilian Integration*

PENDIENTE

**Juliana Xavier**

*Actions of solvable Baumslag-Solitar groups on higher genus surfaces*

Let \(S\) be a closed hyperbolic surface, and let \(f, h: S \to S\) be homeomorphisms such that \(h f h^{-1} = f^n\), for some \( n\geq 1\). Suppose that \(f\) is isotopic to the identity and \(h\) is pseudo-Anosov with stretch factor \(\lambda >1\). We show that \(\lambda > n\) implies \(f= id\).