| 13h30-14h:30 |
Matinê |
Carlos Menino Coton
(UFF) |
Uma variedade exótica que não é folha em qualquer regularidade
|
RESUMO: Presentamos uma variedade que não é difeomorfa a nenhuma folha de nenhuma folheação de codimensão 1 em uma variedade compacta. Essa variedade é homeomorfa mas não difeomorfa ao espaço euclideano de dimensão 4 excluindo um conjunto enumerável (infinito) e discreto de pontos. Este é o primeiro exemplo duma variedade que é homeomorfa a uma folha mas não difeomorfa a nenhuma folha na mínima regularidade razoável para esse problema (a holonomia só é assumida contínua). Em regularidade C2 os autores já mostraram a existência de R4 exóticos que não são difeomorfas a nenhuma folha por causa do Lemma de Kopell de folheações. A variedade será construída colando fins de certas estruturas exóticas de R4 em uma esfera. A prova usará a inter-relação entre a topologia das folhas e a dinâmica transversa sendo a decomposição de Dippolito a ferramenta essencial na nossa análise. Este é um trabalho em conjunto com Paul Schweitzer (PUC-Rio)
|
| 14h40 - 15h:40 |
Palestra1 |
Carlos H. Vasquez
(PUC-Valparaíso, Chile) |
Regularity of Lyapunov exponents |
| RESUMO: Let us consider a C ∞ – one parameter family of C ∞ diffeomorphisms f t , t ∈ 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 μ for every t ∈ I , then the integrated Lyapunov exponent λ ( t ) of f t over E ( t ) is well defined. In this talk we discuss about the regularity of the function λ ( t ) . The explicit expansion of λ ( t ) , involves the regularity of the family, regularity of the splitting and the smooth vector field X on M , which is tangent to the family f − 1 0 ◦ f t in t = 0 . Within the context of dominated splittings. We conjecture that if the 2 bundles of the splitting are of class C α and respectively C β for f 0 , then the integrated Lyapunov exponent λ ( t ) of each bundle has expansion of order ( α + β ) in the parameter t at t = 0 . We prove the conjecture in some cases, using some tools that may have their own interest. Finally, if the time allows it, we discuss about some possible applications of the previous results including the case of variable measure, the abundance of non–uniform hyperbolicity, the abundance of non–absolute continuous central and intermediate foliations, regularity of metric entropy and rigidity of Anosov maps on surfaces. Work in progress joint with Radu Saghin and Pancho Valenzuela-Henríquez. |
| Intervalo para o café (15h:40 - 16h:00) |
| 16h00 - 17h00 |
Palestra 2 |
Felipe Riquelme
(PUC-Valparaíso, Chile) |
Thermodynamical formalism and zero entropy in negative curvature |
| RESUMO: Let X be the unit tangent bundle of a complete negatively curved Riemannian manifold and let ( g t ): X → X be its associated geodesic flow. After the work of R. Bowen and D. Ruelle, it is well know that, if X is compact, then any Hölder-continuous potential F : X ? R admits an equilibrium measure. Moreover, there is a fair enough description of some properties of the pressure map t 7→ P ( tF ) , such as its regularity and its asymptotic behaviour. For non-compact situations, the existence of equilibrium measures has been largely studied with success over the last years. However, when we focus the study on properties of the pressure map, there are still many open questions. In this talk we will consider the particular case of geometrically finite manifolds and positive Hölder- continuous potentials with good behaviour near to infinity. We will prove that the pressure map is C 1 and strictly increasing at least on an unbounded interval containing the interval (0 , ∞ ) . Moreover, on those points where the pressure map is locally constant, there are no equilibrium measures. We also prove that the equilibrium measures for tF do not lose mass when t goes to infinity. We compute the entropy of those limit measures, and as an application, we describe the set of entropies of the geodesic flow with respect to ergodic measures. This is a joint work with Anibal Velozo. |
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