arXiv subjects

Samuel Tapie

Publications and source records attributed to Samuel Tapie.

14 recordsLinked to original sources

arXiv2023

Exotic local limit theorems at the phase transition in free products

We construct random walks on free products of the form Z 3 * Z d , with d = 5 or 6 which are divergent and not spectrally positive recurrent. We then derive a local limit theorem for these random walks, proving that $\mu$ * n (e) $\sim$ CR --n n --5/3 if d = 5 and $\mu$ * n (e) $\sim$ CR --n n --3/2 log(n) --1/2 if d = 6, where $\mu$ * n is the nth convolution power of $\mu$ and R is the inverse of the spectral radius of $\mu$. This disproves a result of Candellero and Gilch [7] and a result of the authors of this paper that was stated in a rst version of [11]. This also shows that the classication of local limit theorems on free products of the form Z d 1 * Z d 2 or more generally on relatively hyperbolic groups with respect to virtually abelian subgroups is incomplete.

math.DS↗

arXiv2023

A local limit theorem for convergent random walks on relatively hyperbolic groups

We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius.When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form $p_n(e, e)\sim CR^{-n}n^{-d/2}$, where $p_n(e, e)$ is the probability of returning to the origin at time $n$, $R$ is the inverse of the spectral radius of the random walk and $d$ is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate.This concludes the classification all possible behaviour for $p_n(e, e)$ on such groups.

math.DS↗

arXiv2020

Pressure at infinity and strong positive recurrence in negative curvature

In the context of geodesic flows of noncompact negatively curved manifolds, we propose three different definitions of entropy and pressure at infinity, through growth of periodic orbits, critical exponents of Poincar\'e series, and entropy (pressure) of invariant measures. We show that these notions coincide. Thanks to these entropy and pressure at infinity, we investigate thoroughly the notion of strong positive recurrence in this geometric context. A potential is said strongly positively recurrent when its pressure at infinity is strictly smaller than the full topological pressure. We show in particular that if a potential is strongly positively recurrent, then it admits a finite Gibbs measure. We also provide easy criteria allowing to build such strong positively recurrent potentials and many examples.

math.DG↗

arXiv2020

Critical exponents of normal subgroups in higher rank

We study the critical exponents of discrete subgroups of a higher rank semi-simple real linear Lie group $G$. Let us fix a Cartan subspace $\mathfrak a\subset \mathfrak g$ of the Lie algebra of $G$. We show that if $Γ< G$ is a discrete group, and $Γ' \triangleleft Γ$ is a Zariski dense normal subgroup, then the limit cones of $Γ$ and $Γ'$ in $\mathfrak a$ coincide. Moreover, for all linear form $ϕ: \mathfrak a\to \mathbb R$ positive on this limit cone, the critical exponents in the direction of $ϕ$ satisfy $\displaystyle δ_ϕ(Γ') \geq \frac 1 2 δ_ϕ(Γ)$. Eventually, we show that if $Γ'\backslash Γ$ is amenable, these critical exponents coincide.

math.DG↗

arXiv2019

Narrow equidistribution and counting of closed geodesics on noncompact manifolds

We prove the equidistribution of (weighted) periodic orbits of the geodesic ow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously known only for geometrically nite manifolds.

math.DS↗

arXiv2019

Discrete geometry and isotropic surfaces

We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n \geq 3$ the piecewise linear isotropic maps can be chosen so that they are piecewise linear isotropic immersions as well. The proofs are obtained using analogies with an infinite dimensional moment map geometry due to Donaldson. As a byproduct of these considerations, we introduce a numerical flow in finite dimension, whose limit provide, from an experimental perspective, many examples of piecewise linear Lagrangian tori in $R^4$. The DMMF program, which is freely available, is based on the Euler method and shows the evolution equation of discrete surfaces in real time, as a movie.

math.DG↗

arXiv2018

Twisted Patterson-Sullivan measures and applications to amenability and coverings

Let $Γ'<Γ$ be two discrete groups acting properly by isometries on a Gromov-hyperbolic space $X$. We prove that their critical exponents coincide if and only if $Γ'$ is co-amenable in $Γ$, under the assumption that the action of $Γ$ on $X$ is strongly positively recurrent, i.e. has a growth gap at infinity. This generalizes all previously known results on this question, which required either $X$ to be the real hyperbolic space and $Γ$ geometrically finite, or $X$ Gromov hyperbolic and $Γ$ cocompact. This result is optimal: we provide several counterexamples when the action is not strongly positively recurrent.

math.GR↗

arXiv2018

$A\_\infty$ weights and compactness of conformal metrics under $L^{n/2}$ curvature bounds

We study sequences of conformal deformations of a smooth closed Riemannian manifold of dimension $n$, assuming uniform volume bounds and $L^{n/2}$ bounds on their scalar curvatures. Singularities may appear in the limit. Nevertheless, we show that under such bounds the underlying metric spaces are pre-compact in the Gromov-Hausdorff topology. Our study is based on the use of $A_\infty$-weights from harmonic analysis, and provides geometric controls on the limit spaces thus obtained. Our techniques also show that any conformal deformation of the Euclidean metric on $R^n$ with infinite volume and finite $L^{n/2}$ norm of the scalar curvature satisfies the Euclidean isoperimetric inequality.

math.DG↗

arXiv2018

Regularity of entropy, geodesic currents and entropy at infinity

In this work, we introduce the notion of entropy at infinity, and define a wide class of noncompact manifolds with negative curvature, those which admit a critical gap between entropy at infinity and topological entropy. We call them strongly positively recurrent manifolds (SPR), and provide many examples. We show that dynamically, they behave as compact manifolds. In particular, they admit a finite measure of maximal entropy. Using the point of view of currents at infinity, we show that on these SPR manifolds the topological entropy of the geodesic flow varies in a C 1 -way along (uniformly) C 1 -perturbations of the metric. This result generalizes former work of Katok (1982) and Katok-Knieper-Weiss (1991) in the compact case.

math.DS↗

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↗

arXiv2011

Dynamics and zeta functions on conformally compact manifolds

In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces, we obtain optimal meromorphic extensions of weighted dynamical zeta functions and asymptotic counting estimates for the number of weighted closed geodesics. A meromorphic extension of the standard dynamical zeta function and the prime orbit theorem follow as corollaries. Finally, we investigate interactions between the dynamics and spectral theory of these spaces.

math.DG↗

arXiv2010

Graphes, moyennabilité et bas du spectre de variétés topologiquement infinies

From a graph $G$ with constant valency $v$ and a (non-compact) manifold $C$ with $v$ boundary components, we build a $G$-periodic manifold $M$. This process gives a class of topologically infinite manifolds which generalizes periodic manifolds and includes all riemannian coverings with finitely generated deck-group. Our main result is that, when the first eigenfunction of $C$ extends to $M$, the bottom of the spectrum of $M$ is equal to $C$'s if and only if the graph $G$ is amenable. When $G$ is not amenable, we control explicitly the gap between these bottom of the spectrum. In particular, we show that if $p : M\ra N$ is a riemannian covering and the metric of $N$ is generic, then $λ_0(M)\geq λ_0(N)$ with equality if and only if the deck-group is amenable. This generalizes a result of R. Brooks.

math.DG↗

arXiv2010

Generic metrics, eigenfunctions and riemannian coverings of non compact manifolds

Let $(M,g)$ be a non-compact riemannian $n$-manifold with bounded geometry at order $k\geq\frac{n}{2}$. We show that if the spectrum of the Laplacian starts with $q+1$ discrete eigenvalues isolated from the essential spectrum, and if the metric is generic for the $\Cl C^{k+2}$-strong topology, then the eigenvalues are distinct and their associated eigenfunctions are Morse. This generalizes to non-compact manifolds some arguments developped by K. Uhlenbeck. We deduce from this result that if $M^n$ has bounded geometry at order $k\geq\frac{n}{2}$ and has an isolated first eigenvalue for its Laplacian, then for any riemannian covering $p : M'\ra M$, we have $λ_0(M) = \sup_D λ_0(D)$, where $D\subset M'$ runs over all connected fundamental domains for $p$, and $λ_0(D)$ is the bottom of the spectrum of $D$ with Neumann boundary conditions.

math.DG↗

arXiv2008

Bas du spectre de surfaces hyperboliques de volume infini

This article presents some methods to control the bottom of the spectrum of the Laplacian $λ_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $λ_0$ of a geometrically finite surface in terms of the geometry of its convex core. We then focus on infinite type periodic hyperbolic surfaces built by gluing copies of a geometrically finite surface with boundary according to the plan of an infinite graph. We control the $λ_0$ of the so-obtained infinite surfaces by constants coming from spectral properties of the building brick and combinatorial datae of the graph. We then use these methods to control the $λ_0$ of two other kind of infinite type hyperbolic surfaces : those admitting a splitting into bounded pieces, and some riemannian coverings.

math.DG↗