arXiv subjects

Pierre Vidotto

Publications and source records attributed to Pierre Vidotto.

2 recordsLinked to original sources

arXiv2017

Counting for some convergent groups

We present examples of geometrically finite manifolds with pinched negative curvature, whose geodesic flow has infinite non-ergodic Bowen-Margulis measure and whose Poincaré series converges at the critical exponent $δ_Γ$. We obtain an explicit asymptotic for their orbital growth function. Namely, for any $α\in ]1, 2[ $ and any slowly varying function $L : \mathbb R\to (0, +\infty)$, we construct $N$-dimensional Hadamard manifolds $(X, g)$ of negative and pinched curvature, whose group of oriented isometries admits convergent geometrically finite subgroups $Γ$ such that, as $R\to +\infty$, $$ N_Γ(R):= \#\left\{γ\in Γ\; ; \; d(o, γ\cdot o)\leq R\right\} \sim C_Γ\frac{L(R)}{R^α} \ e^{δ_ΓR}, $$ for some constant $C_Γ>0$.

math.DS↗

arXiv2017

Ergodic properties of some negatively curved manifolds with infinite measure

Let $M=X/Γ$ be a geometrically finite negatively curved manifold with fundamental group $Γ$ acting on $X$ by isometries. The purpose of this paper is to study the mixing property of the geodesic flow on $T^1M$, the asymptotic equivalent as $R\longrightarrow+\infty$ of the number of closed geodesics on $M$ of length less than $R$ and of the orbital counting function $\sharp\{γ\inΓ |\ d(o,γ.o)\le R\}$. These properties are well known when the Bowen-Margulis measure on $T^1M$ is finite. We consider here divergent Schottky groups whose Bowen-Margulis measure is infinite and ergodic, and we precise these ergodic properties using a suitable symbolic coding.

math.DS↗