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François Ledrappier

Publications and source records attributed to François Ledrappier.

16 recordsLinked to original sources
arXiv2023

Dimension gap and variational principle for Anosov representations

We consider a representation of a finitely generated CAT(--K) group $Γ$ in SL(d, R) that is Zariski dense and k-Anosov for at least two values of k. We exhibit a gap for the Minkowski dimension of minimal sets for the action of $Γ$ on flags spaces. The proof uses a variational principle for the action on partial flags.

math.DS↗
arXiv2023

Exact dimension of dynamical stationary measures

We consider a random walk on $SL_d(\mathbb{R})$ with finite first moment and finite entropy. We show that the distributions of the unstable flag space and of the stable flag space are exact dimensional.

math.DS↗
arXiv2020

Local Limit Theorem in negative curvature

Consider the heat kernel $p(t,x,y)$ on the universal cover $X$ of a Riemannian manifold $M$ of negative curvature. We show the local limit theorem for $p$ : $$\lim_{t \to \infty} t^{3/2}e^{λ_0 t} p(t,x,y)=C(x,y),$$ where $λ_0$ is the bottom of the spectrum of the geometric Laplacian and $C(x,y)$ is a positive function which depends on $x, y \in X$. We also show that the $λ_0$-Martin boundary of $X$ is equal to its topological boundary. The Martin decomposition of $C(x,y)$ gives a family of measures $\{μ^{λ_0}_x \}$ on $\partial \widetilde{M}$. We show that $\{μ^{λ_0}_x \}$ is the unique family minimizing the energy or the Rayleigh quotient of Mohsen. We use the uniform Harnack inequality on the boundary $\partial X$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs-Margulis measures.

math.DS↗
arXiv2019

A family of stable diffusions

Consider a $C^{\infty}$ closed connected Riemannian manifold $(M, g)$ with negative curvature. The unit tangent bundle $SM$ is foliated by the (weak) stable foliation $\mathcal{W}^s$ of the geodesic flow. Let $Δ^s$ be the leafwise Laplacian for $\mathcal{W}^s$ and let $\overline{X}$ be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each $λ$, the operator $\mathcal{L}_λ:=Δ^s+λ\overline{X}$ generates a diffusion for $\mathcal{W}^s$. We show that, as $λ\to -\infty$, the unique stationary probability measure for the leafwise diffusion of $\mathcal{L}_λ$ converges to the normalized Lebesgue measure on $SM$.

math.DS↗
arXiv2018

The regularity of the linear drift in negatively curved spaces

We show the linear drift of the Brownian motion on the universal cover of a closed connected Riemannian manifold is $C^{k-2}$ differentiable along any $C^{k}$ curve in the manifold of $C^k$ metrics with negative sectional curvature. We also show that the stochastic entropy of the Brownian motion is $C^1$ differentiable along any $C^{3}$ curve of $C^3$ metrics with negative sectional curvature. We formulate the first derivatives of the linear drift and entropy, respectively, and show they are critical at locally symmetric metrics.

math.DS↗
arXiv2015

Regularity of the drift and entropy of random walks on groups

Random walks on a group $G$ model many natural phenomena. A random walk is defined by a probability measure $p$ on $G$. We are interested in asymptotic properties of the random walks and in particular in the linear drift and the asymptotic entropy. If the geometry of the group is rich, then these numbers are both positive and the way of dependence on $p$ is itself a property of $G$. In this note, we review recent results about the regularity of the drift and the entropy for free groups, free products and hyperbolic groups.

math.PR↗
arXiv2015

Ergodic properties of equilibrium measures for smooth three dimensional flows

Let $\{T^t\}$ be a smooth flow with positive speed and positive topological entropy on a compact smooth three dimensional manifold, and let $μ$ be an ergodic measure of maximal entropy. We show that either $\{T^t\}$ is Bernoulli, or $\{T^t\}$ is isomorphic to the product of a Bernoulli flow and a rotational flow. Applications are given to Reeb flows.

math.DS↗
arXiv2012

Besicovitch-Federer projection theorem and geodesic flows on Riemann surfaces

We extend the Besicovitch-Federer projection theorem to transversal families of mappings. As an application we show that on a certain class of Riemann surfaces with constant negative curvature and with boundary, there exist natural 2-dimensional measures invariant under the geodesic flow having 2-dimensional supports such that their projections to the base manifold are 2-dimensional but the supports of the projections are Lebesgue negligible. In particular, the union of complete geodesics has Hausdorff dimension 2 and is Lebesgue negligible.

math.CA↗
arXiv2011

Noncommutative Ergodic Theorems

We present recent results about the asymptotic behavior of ergodic products of isometries of a metric space X. If we assume that the displacement is integrable, then either there is a sublinear diffusion or there is, for almost every trajectory in X, a preferred direction at the boundary. We discuss the precise statement when X is a proper metric space and compare it with classical ergodic theorems. Applications are given to ergodic theorems for nonintegrable functions, random walks on groups and Brownian motion on covering manifolds.

math.DS↗
arXiv2009

Linear drift and entropy for regular covers

We consider a regular Riemannian cover $\M$ of a compact Riemannian manifold. The linear drift $\ell$ and the Kaimanovich entropy $h$ are geometric invariants defined by asymptotic properties of the Brownian motion on $\M$. We show that $\ell^2 \leq h$.

math.DS↗
arXiv2006

On laws of large numbers for random walks

We prove a general noncommutative law of large numbers. This applies in particular to random walks on any locally finite homogeneous graph, as well as to Brownian motion on Riemannian manifolds which admit a compact quotient. It also generalizes Oseledec's multiplicative ergodic theorem. In addition, we show that $ε$-shadows of any ballistic random walk with finite moment on any group eventually intersect. Some related results concerning Coxeter groups and mapping class groups are recorded in the last section.

math.PR↗