Journée GAD 2019 |

Résumé
Orateurs/programme :
Abstract: A family of geodesics emanating from a point \(p\) on a manifold \(M\) may come to focus due to the curvature of \(M\). The focal set is sometimes referred to as the conjugate locus, and it can be very complex. Nonetheless we can show a simple relationship between the topological and geometrical properties of the conjugate locus on convex surfaces, and continue to describe the conjugate locus in manifolds of dimension 3. We also consider the integrability of geodesic flows on certain simple surfaces, presenting examples of surfaces with non-integrable geodesic flow as well as constructing new examples with integrable geodesic flow.
Let (M,g) = (X/\Gamma, g) be a negatively curved manifold with universal cover X. The critical exponent \delta_\Gamma(g) is a number which measures the topological and geometrical complexity of (M,g). For locally symmetric manifolds, it is strongly related to the spectrum of the Laplacian. If N = M/ \Gamma’ is a covering of M, the critical exponent of the subgroup \Gamma'<\Gamma is hence smaller than the critical exponent of \Gamma. When does equality occurs ? It was shown in the 1980’s by Brooks that if M is a convex co-compact hyperbolic manifold and \Gamma’ is a normal subgroup of \Gamma, then equality occurs if and only if \Gamma/\Gamma’ is amenable. At the same time, Cohen and Grigorchuk showed an analogous result when \Gamma is a free group acting on its Cayley graph. When the action of M /\Gamma is not convex cocompact, showing that equality of critical exponents is equivalent to the amenability of \Gamma/\Gamma’ requires an additional assumption: a “critical gap at infinity”. I will explain how under this (optimal) assumption, we can generalize the result of Brooks to all non-compact negatively curved manifolds. Joint work with R. Coulon, R. Dougall and B. Schapira
I will discuss a general ergodic theorem for compositions of randomly selected transformations, which is a setting that generalizes the classical law of large numbers and the iteration of just one single map. The theorem will be given in terms of metric functionals, which is an extension of the concept of horofunction of Busemann-Gromov. The metric functionals provide a weak topology and compactness to metric spaces, and as tool allows for some very general statements that can be viewed a bit in parallel to linear functional analysis. The list of applications is rather long. The proof of the main theorem is based on a substantial refinement of Kingman’s subadditive ergodic theorem. Joint work with S. Gouezel.
Résumé : The Ruelle resonances of a dynamical system are spectral characteristics of the system, describing the precise asymptotics of correlations. While one can usually show their existence by abstract spectral arguments, they are most of the time not computable. I will explain that, in the case of linear pseudo-Anosov maps, one can describe them explicitly in terms of the action of the pseudo-Anosov on cohomology. Joint with Frédéric Faure and Erwan Lanneau. |