Palestrantes
Title: Ruelle resonances for the geodesic flow of noncompact negatively
curved manifolds.
Abstract: We are interested in the stochastic properties of the geodesic flow of a noncompact negatively curved manifold M. Let m_F be a Gibbs measure associated with a smooth potential F on the unit tangent bundle T^1M. Under a dynamical assumption, called strong positive recurrence, saying that a pressure at infinity \delta^F_\infty is strictly smaller than the topological pressure \delta^F, this measure m_F is finite, mixing, and admits Ruelle resonances at least up to \delta^F_\infty-\delta^F<0. This is a joint work with Sébastien Gouezel and Samuel Tapie. In my talk, I will explain and motivate this abstract, and give some flavour of the proof.
Title: Rigidity of u-Gibbs measures for perturbations of conservative Anosov diffeomorphisms of T^3.
Abstract: Let f be a partially hyperbolic diffeomorphism with a three way splitting E^s + E^c + E^u on T^3. In this context, measures absolutely continuous with respect to the Lebesgue length along the strong unstable direction E^u play a prominent role. They are called u-Gibbs measures and their existence was established by Pesin and Sinai, in the eighties. In more specific situations, like when the center is mostly contracting, u-Gibbs measures are the physical measures of the system and thus capture the asymptotic statistical behaviour of almost every initial trajectory and, moreover, there is a finite number of them. The goal of this talk is to study u-Gibbs measures for C^2 partially hyperbolic diffeomorphisms with a three way splitting E^s + E^c + E^u which belong to a C^1 neighborhood of conservative Anosov diffeomorphisms of T^3, thus presenting an expanding center E^c and a two dimensional unstable direction. We show that if the strong bundles E^s and E^u are not jointly integrable then every u-Gibbs measure is absolutely continuous with respect to the Lebesgue area along the unstable direction E^c + E^u and therefore is an SRB measure, the same conclusion as in the mostly contracting case. In particular, we deduce uniqueness of u-Gibbs measures. Our proof fits well within the context of measure rigidity results in dynamics. Indeed, to deduce information on the geometry of the measure along the two dimensional unstable direction from information along the strong unstable one dimensional direction we employ a version of Benoist-Quint's exponential drift argument, as given (mostly) by Eskin-Lindenstrauss and Eskin-Mirzakhani, and also some ideas from Brown-Hertz. This is joint work with Sébastien Alvarez, Martin Leguil and Davi Obata.
Title: Building non-hyperbolic measures.
Abstract: In a sequence of joint works with Jairo Bochi, Lorenzo Díaz and myself, we build in a robust way non-hyperbolic ergodic measures, each time with more requirements on the support and entropy. Any new property we got needed a new principle. I will try to present (some of) these principles.
Title: BBB: before blender and beyond.
Abstract: We will chat about the blender's influence in dynamical systems; a toy, a tool, that it's becoming a concept.
Title: Hyperbolicity of renormalization for bi-cubic circle maps with bounded combinatorics.
Abstract: Renormalization theory has been developed for real-analytic circle maps with one non-degenerate critical point. In this talk, we study real-analytic circle maps with two cubic critical points and with bounded type rotation number. We define a suitable functional space where these maps embed and we show that, in this functional space, renormalization has a hyperbolic attractor with codimension-two stable foliation. This is joint work with Michael Yampolsky.
Title: The fundamental inequality for cocompact Fuchsian groups.
Abstract: A recurring question in the theory of random walks on hyperbolic spaces asks whether the hitting (harmonic) measures can coincide with measures of geometric origin, such as the Lebesgue measure. This is also related to the inequality between entropy and drift. For finitely supported random walks on cocompact Fuchsian groups with symmetric fundamental domain, we prove that the hitting measure is singular with respect to Lebesgue measure; moreover, its Hausdorff dimension is strictly less than 1. Along the way, we prove a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups. Joint with P. Kosenko.
Title: Measures of maximal u-entropy for maps that factor over Anosov.
Abstract: This is a joint work with Raul Ures, Marcelo Viana and Fan Yang. We construct measures of maximal u-entropy for any partially hyperbolic diffeomorphism that factors over an Anosov torus automorphism and has mostly contracting center direction. The space of such measures has a finite dimension, and its extreme points are ergodic measures with pairwise disjoint supports.
Title: Porcupines and other prickly animals.
Abstract: I will tell you about some results of Lorenzo on three dimensional systems with invariant central manifold, with stress put on their applicability in many areas of dynamical systems.
Title: Renormalization for interval exchange transformations with flips.
Abstract: Renormalization is a strategy that permits to replace a dynamical system by its renormalized version which is, for example, a first-return map on some open subset of the phase space or an acceleration of the parametrization of the flow. This strategy has been extremely useful in the last 50 years for understanding of the dynamics in different contexts such as complex dynamics (form of the Mandelbrot set), translation surfaces (Teichmüller flow), 1-dimensional dynamics and many others. I will consider in detail one specific example where the renormalization techniques work for a family of interval exchange transformations with flips, giving unexpected consequences on the dynamics and describing fully the symbolic dynamics.
Title: Heterogeneously coupled maps: hub dynamics and emergence.
Abstract: We study the dynamics of Heterogeneously Coupled Maps (HCM). Such systems are determined by a network with heterogeneous degrees. Some nodes, called hubs, are very well connected while most nodes interact with few others. The local dynamics on each node is chaotic, coupled with other nodes according to the network structure. Such
high-dimensional systems are hard to understand in full, nevertheless we are able to describe the system over exponentially large time scales. In particular, we show that the dynamics of hub nodes can be very well approximated by a low-dimensional system. This allows us to establish the emergence of macroscopic behaviour such as coherence of dynamics among hubs of the same connectivity layer (i.e. with the same number of connections), and chaotic behaviour of the poorly connected nodes. The HCMs we study provide a paradigm to explain why and how the dynamics of the network can change across layers.
Title: On equilibrium states for a family of partially hyperbolic horseshoes.
Abstract: Invariant measures that maximize the topological pressure of a dynamical system, called equilibrium states, provide relevant information about the statistical behavior of typical orbits. Despite many advances in the area, the theory of equilibrium states in the partially hyperbolic context is still far from being well understood. In this talk we discuss the problem of equilibrium states for the family of partially hyperbolic horseshoes introduced by Díaz L., Horita, V., Rios, I. and Sambarino, M.; We define a projection map associated to the horseshoe and we prove that the transfer operator of this map has the spectral gap property in a space of Hölder continuous observables. From this result, we derive uniqueness as well as good statistical and topological properties for the equilibrium state of the horseshoe. This is a joint work with J. Siqueira (UFRJ).
Title: Bounded deviation behavior for dynamics with a certain rotation set.
Abstract: For a non-wandering homeomorphism of the two torus, if the rotation set of a lifted dynamics F, is a line segment from (0,0) to some totally irrational vector (\alpha,\beta), then we show F exhibits bounded displacement along the direction -(\alpha, \beta). This result is shown via some new results on the existence of non-contractible periodic orbits for surface dynamics. This is joint work with Fabio Tal.