# Dynamical Day July 12th, 2018

Facultad de Matemáticas, UC.Tha aim is of this is meeting is twofold, on the one hand we will gather a group of dynamicists to discuss new developments in the area and on the other we will have the Master´s defense of Renato Velozo.

Jairo Bochi

Godofredo Iommi

Jan Kiwi

Mario Ponce

**1. Carlos Vásquez (Pontificia Universidad Católica de Valpraíaso, Chile).**

**Title: **Regularity of Lyapunov Exponents**Abstract: **In this work, we consider a $C^infty$–,one parameter family of $C^{infty}$ diffeomorphisms $f_t$, $tin I$, defined on a compact orientable Riemannian manifold $M$. If the family admits a $Df_t$–invariant subbundle $E_t$ and an invariant probability measure $mu$ for every $tin I$, then the integrated Lyapunov exponent $lambda(t)$ of $f_t$ over $E_t$ is well defined. We discuss about conditions for the differentiability of $lambda(t)$. Work in progress joint with Radu Saghin and Pancho Valenzuela-Henr´iquez.

**2. **Umberto Hryniewicz **(Universidade Federal do Rio de Janeiro, Brazil).**

**Title**: Existence of global cross-sections: from Schwartzman cycles to holomorphic curves

**Abstract: **The notion of a global section for a flow in dimension three goes back to the work of Poincare in Celestial Mechanics. During the second half of the XX century, initiated with the work of Sol Schwartzman, the construction of global cross-sections was organized as the study of linking properties of invariant measures. Fundamental contributions were given by Fried and Sullivan. Recently Ghys introduced the quadratic linking form and the notion of right-handed vector fields, and studied Lorentz knots. He used Schwartzman-Fried-Sullivan theory to investigate knot types of periodic orbits of right-handed flows and made the following statement: any collection of such orbits is a fibered link that binds an open book decomposition whose pages are global cross-sections. In this talk I would like to explain how holomorphic curves can be used to improve the abstract results from Schwartzman-Fried-Sullivan theory to obtain statements about (large) classes of Reeb flows that require few assumptions.

3.** Renato Velozo (Ponitificia Universidad Católica de Chile, Chile).**

Tilte: Characterization of uniform hyperbolcity for fiber bunched cocycles

**Abstract: **We prove a characterization of uniform hyperbolicity for fiberbunched cocycles. Specifically, we show that the existence of a uniform gap between the Lyapunov exponents of a fiber-bunched SLp2, Rq-cocycle defined over a subshift of finite type or an Anosov diffeomorphism implies uniform hyperbolicity. In addition, we construct an α-H¨older cocycle which has uniform gap between the Lyapunov exponents, but it is not uniformly hyperbolic.

This event is supported by CONICYT through the proyecto Anillo ACT172001, «New trends in Ergodic Theory» and by the Facultad de Matremáticas, Pontificia Universidad Católica de Chile