Orbital functions and heat kernels of Kleinian groups
We study orbital functions associated to Kleinian groups through the heat kernel approach developed in \cite{artmoiheatcounting1}.
math.DG↗
arXiv subjects
Adrien Boulanger
Publications and source records attributed to Adrien Boulanger.
The Dirichlet random walk
In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We prove a theorem à la Kesten for such a process: the escape rate of the random walk is positive if and only if the cover is non amenable. We also investigate more in details the case where the deck group is Gromov hyperbolic, showing the almost sure convergence to the boundary of the trajectory as well as a central limit theorem for the escape rate
math.PR↗
Closed geodesics in dilation surfaces
We prove that directions of closed geodesics in every dilation surface form a dense subset of the circle. The proof draws on a study of the degenerations of the Delaunay triangulation of dilation surfaces under the action of Teichm\"{u}ller flow in the moduli space.
math.GT↗
Large deviations for random walks on Gromov-hyperbolic spaces
Let $Γ$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $μ$ a probability measure on $Γ$ whose support generates a non-elementary subsemigroup. Under the assumption that $μ$ has a finite exponential moment, we establish large deviations results for the distance and the translation length of a random walk with driving measure $μ$. From our results, we deduce a special case of a conjecture regarding large deviations of spectral radii of random matrix products.
math.PR↗
A Cheeger-like inequality for coexact 1-forms
We state and prove a Cheeger-like inequality for coexact 1-forms on closed orientable Riemannian manifolds.
math.DG↗
A stochastic approach to counting problems
We study orbital functions associated to finitely generated geometrically infinite Kleinian groups acting on the hyperbolic space $\mathbb{H}^3$, developing a new method based on the use of the Brownian motion. On the way, we give some estimates of the orbital function associated to nilpotent covers of compact hyperbolic manifolds, partially answering a question asked by M. Pollicott to the author.
math.DG↗
Large deviations and rate of escape for hyperbolic random walks
Let $Γ$ be a countable group acting on a geodesic hyperbolic metric space $X$ and $μ$ a probability measure on $Γ$ which generates a non elementary semi-group. Under the necessary assumption that $μ$ has a finite exponential moment, we establish large deviations results for the distance of a random walk with driving measure $μ$.
math.PR↗
$\mathrm{SL}_2(\mathbb{R})$-dynamics on the moduli space of one-holed tori
We study the $\mathrm{SL}_2(\mathbb{R})$-action on the moduli space of (triangulable) dilation tori with one boundary component. We prove that every orbit is either closed or dense, and that every orbit of the Teichmuller flow escapes to infinity.
math.DS↗
A spectral approach to the linking number in the 3-torus
Given a closed Riemannian manifold and a pair of multi-curves in it, we give a formula relating the linking number of the later to the spectral theory of the Laplace operator acting on differential one forms. As an application, we compute the linking number of any two multi-geodesics of the flat torus of dimension 3, generalising a result of P. Dehornoy.
math.GT↗
Cascades in the dynamics of affine interval exchange transformations
We describe in this article the dynamics of a $1$-parameter family of affine interval exchange transformations. It amounts to studying the directional foliations of a particular affine surface, the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial, but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. s study is achieved through the analysis the dynamics of the Veech group of this surface combined a modified version of Rauzy induction in the context of affine interval exchange transformations.
math.DS↗