Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps

Abstract

Abstract. -We consider a large class of non compact hyperbolic manifolds M = H n /Γ with cusps and we prove that the winding process (Yt) generated by a closed 1-form supported on a neighborhood of a cusp C, satisfies a limit theorem, with an asymptotic stable law and a renormalising factor depending only on the rank of the cusp C and the Poincaré exponent δ of Γ. No assumption on the value of δ is required and this theorem generalises previous results due to Y.

Spectral gap properties for linear random walks and Pareto’s asymptotics for affine stochastic recursions

“…We observe that, more generally, the homogeneous behaviour at infinity of certain invariant measures is of interest for various questions in Probability Theory and Mathematical Physics (see [10], [11], [12], [13], [35], [47]) but also in some geometrical questions such as dynamical excursions of geodesic flow and winding around cusps in hyperbolic manifolds (see [1], [44], [49]), or analysis of the H-space (V, ρ) as a λ-boundary and its dynamical consequences (see [3], [16], [17], [30]). …”

“…If s = 0, P s reduces to the convolution operator by µ on P and convergence to the unique µ-stationary measure ν 0 = ν was studied in [17] using proximality of the Taction on P d− 1 . In this case, spectral gap properties for P z , if Rez= s is small, were first proved in [40] using the simplicity of the dominant µ-Lyapunov exponent (see [28]).…”

“…) is compact and convex, Schauder-Tychonov theorem implies the existence of k > 0 and (1). For such a σ, the equation…”

“…Proof of Theorem 2.6: As in the proof of Lemma 2.8, we consider the non linear operator (1) . The same argument gives the existence of …”

“…For information on the role of spectral gap properties in limit theorems for Probability theory and Ergodic theory we refer to [1], [4], [6], [18], [21], [27], [28], [40]. For information on products of random matrices we refer to [4], [6], [16], [24].…”