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Timothée Bénard

Publications and source records attributed to Timothée Bénard.

At least 19 recordsLinked to original sources
arXiv2026

Proof of the singularity conjecture for discrete subgroups of $\mathrm{SL}_2(\mathbb{R})$

Let $μ$ be a finitely supported probability measure on $\mathrm{SL}_2(\mathbb{R})$ whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on $\mathbb{P}^1(\mathbb{R})$ is singular with respect to the $\mathrm{SO}(2)$-invariant probability measure. The proof compares winding variances on finite covers of a closed hyperbolic surface and uses a topological obstruction.

math.DS↗
arXiv2026

Multislicing and effective equidistribution for random walks on some homogeneous spaces

We consider a random walk on a homogeneous space $G/Λ$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $Λ$ is a lattice. The walk is driven by a probability measure $μ$ on $G$ whose support generates a Zariski-dense subgroup. We show that for every starting point $x \in G/Λ$ which is not trapped in a finite $μ$-invariant set, the $n$-step distribution $μ^{*n}*δ_{x}$ of the walk equidistributes toward the Haar measure. Moreover, under arithmetic assumptions on the pair $(Λ, μ)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that $x$ may be high in a cusp or close to a finite orbit. Our approach is substantially different from that of Benoist-Quint, whose equidistribution statements only hold in Cesàro average and are not quantitative, that of Bourgain-Furman-Lindenstrauss-Mozes concerning the torus case, and that of Lindenstrauss-Mohammadi-Wang and Yang about the analogous problem for unipotent flows. A key new feature of our proof is the use of a new phenomenon which we call multislicing. The latter is a generalization of the discretized projection theorems à la Bourgain and we believe it presents independent interest.

math.DS↗
arXiv2026

Effective equidistribution of random walks on simple homogeneous spaces

We consider a random walk on a homogeneous space $G/Λ$ where $G$ is a non-compact simple Lie group and $Λ$ is a lattice. The walk is driven by a probability measure $μ$ on $G$ whose support generates a Zariski-dense subgroup. We show that the random walk equidistributes towards the Haar measure unless it is trapped in a finite $μ$-invariant set. Moreover, under arithmetic assumptions on the pair $(Λ, μ)$, we show the convergence occurs at an exponential rate, tempered by the obstructions that the starting point may be high in a cusp or close to a finite orbit. The main challenge is to show that the dimensional properties of a given probability distribution on $G/Λ$ improve under convolution by $μ$. For this, we develop a new method, which combines a dimensional interpolation result and a dimensional increase alternative. This approach allows us to bypass inherent geometric obstructions. To show dimensional interpolation, we establish a general subcritical projection theorem under optimal non-concentration assumptions on the projector, and a corresponding submodular inequality for irreducible representations which allows its application to random walks. Both are of independent interest. The dimensional increase alternative aligns with the spirit of Bourgain's projection theorem. It is fine-tuned for random walks and has the advantage of being valid in situations lacking transversality.

math.DS↗
arXiv2026

Rigidity and equidistribution of random walks by diffeomorphisms near the conservative regime

We consider a random walk on a closed manifold $M$ driven by a probability measure $μ$ on the space of $C^2$ diffeomorphisms. Provided $μ$ has compact support, satisfies certain gap and pinching conditions, and is weak-$*$ close to a volume-preserving measure, we prove that $M$ carries a unique atom-free stationary probability measure $Υ_μ$. This measure has full Frostman dimension and coincides with volume in the volume-preserving setting. Moreover, for every $x\in M$, the $n$-step distribution $μ^{*n} * δ_x$ converges to $Υ_μ$ unless $x$ is contained in a finite $μ$-invariant set. Our result applies to a variety of situations, including bi-expanding random walks on surfaces, non-linear perturbations of Zariski-dense random walks on the torus $\mathbb{T}^d$, on cocompact lattice quotients of $\mathrm{SO}(2,1)$ and $\mathrm{SO}(3,1)$, and on the sphere $\mathbb{S}^d$.

math.DS↗
arXiv2026

Khintchine dichotomy and Schmidt estimates for self-similar measures on $\mathbb{R}^d$

We extend the classical theorems of Khintchine and Schmidt in metric Diophantine approximation to the context of self-similar measures on $\mathbb{R}^d$. For this, we establish effective equidistribution of associated random walks on $\text{SL}_{d+1}(\mathbb{R})/\text{SL}_{d+1}(\mathbb{Z})$. This generalizes our previous work which requires $d=1$ and restricts Schmidt-type counting estimates to approximation functions which decay fast enough. Novel techniques include a bootstrap scheme for the associated random walks despite algebraic obstructions, and a refined treatment of Dani's correspondence. Along the way, we also establish non-concentration properties of self-similar measures near algebraic subvarieties of $\mathbb{R}^d$.

math.DS↗
arXiv2025

Khintchine dichotomy for self-similar measures

We establish the analogue of Khintchine's theorem for all self-similar probability measures on the real line. When specified to the case of the Hausdorff measure on the middle-thirds Cantor set, the result is already new and provides an answer to an old question of Mahler. The proof consists in showing effective equidistribution in law of expanding upper-triangular random walks on $\text{SL}_{2}(\mathbb{R})/\text{SL}_{2}(\mathbb{Z})$, a result of independent interest.

math.DS↗
arXiv2025

Noise stability on hyperbolic groups

We show that symmetric random walks on non-elementary hyperbolic groups with non-zero homomorphisms into the reals are noise stable at linear scale under finite exponential moment condition.

math.PR↗
arXiv2024

Winding of geodesic rays chosen by a harmonic measure

We prove limit theorems for the homological winding of geodesic rays distributed via a harmonic measure on a Gromov hyperbolic space. We obtain applications to the inverse problem for the harmonic measure, and winding statistics for Patterson-Sullivan measures.

math.DS↗
arXiv2024

The central limit theorem on nilpotent Lie groups

We formulate and establish the central limit theorem for products of i.i.d. random variables on arbitrary simply connected nilpotent Lie groups, allowing a possible bias. Two new phenomena arise in the presence of a bias: (a) the walk spreads out at a higher rate in the ambient group, (b) the limiting hypoelliptic diffusion process may not have full support. We study the limiting distribution, prove estimates on its density and describe its support. We also establish corresponding Berry-Esseen bounds under optimal moment assumptions, as well as an analogue of Donsker's invariance principle. Various examples of nilpotent Lie groups are treated in detail showing the variety of different behaviours. We also obtain a characterization of when the limiting distribution is an ordinary gaussian and answer a question of Tutubalin from the 1960s regarding asymptotically close distributions on nilpotent Lie groups.

math.PR↗
arXiv2023

Local limit theorems for random walks on nilpotent Lie groups

We establish the (non-lattice) local limit theorem for products of i.i.d. random variables on an arbitrary simply connected nilpotent Lie group $G$, where the variables are allowed to be non-centered. Our result also improves on the known centered case by proving uniformity for two-sided moderate deviations and allowing measures with a moment of order $2(\dim G)^2$ without further regularity assumptions. As applications we establish a Ratner-type equidistribution theorem for unipotent walks on homogeneous spaces and obtain a new proof of the Choquet-Deny property in our setting.

math.PR↗
arXiv2023

Exponential multiple mixing for commuting automorphisms of a nilmanifold

Let $l\in \mathbb{N}_{\geq 1}$ and $α: \mathbb{Z}^l\rightarrow \text{Aut}(\mathscr{N})$ be an action of $\mathbb{Z}^l$ by automorphisms on a compact nilmanifold $\mathscr{N}$. We assume the action of every $α(z)$ is ergodic for $z\in \mathbb{Z}^l\smallsetminus\{0\}$ and show that $α$ satisfies exponential $n$-mixing for any integer $n\geq 2$. This extends results of Gorodnik and Spatzier [Acta Math., 215 (2015)].

math.DS↗
arXiv2022

Some asymptotic properties of random walks on homogeneous spaces

Let $G$ be a connected semisimple real Lie group with finite center, and $μ$ a probability measure on $G$ whose support generates a Zariski-dense subgroup of $G$. We consider the right $μ$-random walk on $G$ and show that each random trajectory spends most of its time at bounded distance of a well-chosen Weyl chamber. We infer that if $G$ has rank one, and $μ$ has a finite first moment, then for any discrete subgroup $Λ\subseteq G$, the $μ$-walk and the geodesic flow on $Λ\backslash G$ are either both transient, or both recurrent and ergodic, thus extending a well known theorem due to Hopf-Tsuji-Sullivan-Kaimanovich dealing with the Brownian motion.

math.DS↗
arXiv2022

Transience in law for symmetric random walks in infinite measure

We consider a random walk on a second countable locally compact topological space endowed with an invariant Radon measure. We show that if the walk is symmetric and if every subset which is invariant by the walk has zero or infinite measure, then one has escape of mass for almost every starting point. We then apply this result in the context of homogeneous random walks on infinite volume spaces, and deduce a converse to Eskin-Margulis recurrence theorem.

math.DS↗
arXiv2022

Random walks with bounded first moment on finite-volume spaces

Let $G$ be a real Lie group, $Λ\leq G$ a lattice, and $Ω=G/Λ$. We study the equidistribution properties of the left random walk on $Ω$ induced by a probability measure $μ$ on $G$. It is assumed that $μ$ has a finite first moment, and that the Zariski closure of the group generated by the support of $μ$ in the adjoint representation is semisimple without compact factors. We show that for every starting point $x\in Ω$, the $μ$-walk with origin $x$ has no escape of mass, and equidistributes in Cesàro averages toward some homogeneous measure. This extends several fundamental results due to Benoist-Quint and Eskin-Margulis for walks with finite exponential moment.

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arXiv2021

Drift of random walks on abelian covers of finite volume homogeneous spaces

Let $G$ be a connected simple real Lie group, $Λ_{0}\subseteq G$ a lattice and $Λ\unlhd Λ_{0}$ a normal subgroup such that $Λ_{0}/Λ\simeq \mathbb{Z}^d$. We study the drift of a random walk on the $\mathbb{Z}^d$-cover $Λ\backslash G$ of the finite volume homogeneous space $Λ_{0}\backslash G$. This walk is defined by a Zariski-dense compactly supported probability measure $μ$ on $G$. We first assume the covering map $Λ\backslash G\rightarrow Λ_{0}\backslash G$ does not unfold any cusp of $Λ_{0}\backslash G$ and compute the drift at \emph{every} starting point. Then we remove this assumption and describe the drift almost everywhere. The case of hyperbolic manifolds of dimension 2 stands out with non-converging type behaviors. The recurrence of the trajectories is also characterized in this context.

math.DS↗
arXiv2021

Equidistribution of mass for random processes on finite-volume spaces

Let $G$ be a real Lie group, $Λ\subseteq G$ a lattice, and $X=G/Λ$. We fix a probability measure $μ$ on $G$ and consider the left random walk induced on $X$. It is assumed that $μ$ is aperiodic, has a finite first moment, spans a semisimple algebraic group without compact factors, and has two non mutually singular convolution powers. We show that for every starting point $x\in X$, the $n$-th step distribution $μ^n*δ_{x}$ of the walk weak-$\ast$ converges toward some homogeneous probability measure on $X$.

math.PR↗