arXiv subjects
Publications and source records attributed to Abdelhamid Amroun.
We prove a central limit theorem for a class of Hölder continuous cocycles with an application to stricly convex and irreducible rational representations of hyperbolic groups, introduced by Sambarino [Quantitative properties of convexe representations. Comment. Math. Helv 89 (2014), 443-488].
Let $ρ: Γ\longrightarrow G$ be a Zariski dense irreducible convex representation of the hyperbolic group $Γ$, where G is a connected real semisimple algebraic Lie group. We establish a central limit type theorem for the periods of the representation $ρ$.
We consider the geodesic flow of a compact connected rank 1 surface. We prove a formula for the topological pressure as the exponential growth rate of rank 1 periodic geodesics generalizing a previous result of K. Gelfert and B. Schapira [12].
We give a general method on the way of approximating equilibrium states for a dynamical system of a compact metric space.
We give a formula for the topological pressure of the geodesic flow of a compact rank 1 manifold in terms of the growth of the number of closed hyperbolic (rank 1) geodesics. We derive an equidistribution result for these geodesics with respect to equilibrium states. This generalize partially a result of G. Knieper \cite{kni} to non constant potentiels.
We prove an equidistribution result for $C^{\infty}$ maps with respect to equilibrium states. We apply the result to the time-one map of the geodesic flow of a closed smooth Riemannian manifold.
Using the works of Mañé \cite{Ma} and Paternain \cite{Pat} we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a $\mathcal{C}^{\infty}$ Riemannian metric. We prove large deviations lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.