In this work, we introduce a notion of entropy at infinity for the geodesic flow of negatively curved manifolds. We introduce the class of noncompact manifolds which admit a critical gap between entropy at infinity and topological entropy. We call them strongly positively recurrent manifolds (SPR), and provide many examples. We show that dynamically, they behave as compact manifolds. In particular, they admit a finite measure of maximal entropy.Using the point of view of currents at infinity, we show that on these SPR manifolds the topological entropy of the geodesic flow varies in a C 1 -way along (uniformly) C 1 -perturbations of the metric. This result generalizes former work of Katok (1982) and Katok-Knieper-Weiss (1991) in the compact case. RésuméDans ce travail, nous introduisons une notion d'entropie à l'infini pour les flots géodésiques des variétés à courbure négative. Nous introduisons la classe des variétés, dites fortement positivement récurrentes (SPR), dont l'entropie à l'infini est strictement inférieure à l'entropie topologique. Nous donnons de nombreux exemples de telles variétés. Nous montrons que d'un point de vue dynamique, ces variétés ressemblent à des variétés compactes. En particulier, elles admettent une mesure finie maximisant l'entropie.A l'aide du point de vue des courants à l'infini, nous montrons que sur ces variétés SPR, l'entropie topologique varie de manière C 1 le long de perturbations C 1 uniformes de la métrique. Ceci généralise des résultats passés de Katok (1982) et Katok-Knieper-Weiss (1991 dans le cas compact.

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We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius. When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form pn(e, e) ∼ CR −n n −d/2 , where pn(e, e) is the probability of returning to the origin at time n, R is the inverse of the spectral radius of the random walk and d is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate. This concludes the classification all possible behaviour for pn(e, e) on such groups.

We introduce two versions of the Yamabe flow which preserve negative scalar-curvature bounds. First we show existence and smooth convergence of solutions to these flows. We then show that a metric with negative scalar curvature is controlled by the Yamabe metrics in the same conformal class with constant extremal scalar curvatures. This implies that the volume entropy of our original metric is controlled by the entropies of these Yamabe metrics. We eventually use these Yamabe flows to prove an entropy-rigidity result: when the Yamabe metric has negative sectional curvature, the entropy of a metric in the same conformal class is extremal if and only if the metric has constant extremal scalar curvature.