arXiv subjects

Benoît Kloeckner

Publications and source records attributed to Benoît Kloeckner.

At least 19 recordsLinked to original sources

arXiv2025

Effective perturbation theory for linear operators

We propose a new approach to the spectral theory of perturbed linear operators , in the case of a simple isolated eigenvalue. We obtain two kind of results: ''radius bounds'' which ensure perturbation theory applies for perturbations up to an explicit size, and ''regularity bounds'' which control the variations of eigendata to any order. Our method is based on the Implicit Function Theorem and proceeds by establishing differential inequalities on two natural quantities: the norm of the projection to the eigendirection, and the norm of the reduced resolvent. We obtain completely explicit results without any assumption on the underlying Banach space. In companion articles, on the one hand we apply the regularity bounds to Markov chains, obtaining non-asymptotic concentration and Berry-Ess{é}en inequalities with explicit constants, and on the other hand we apply the radius bounds to transfer operator of intermittent maps, obtaining explicit high-temperature regimes where a spectral gap occurs.

math.SP↗

arXiv2023

Mixing speed and stability of SRB measures through optimal transportation

It is well-known that the SRB measure of a $C^{1+α}$ Anosov diffeomorphism has exponential decay of correlations with respect to H{ö}lder-continuous observables. We propose a new approach to this phenomenon, based on optimal transport. More precisely, we define a space of measures having absolutely continuous disintegrations with respect to some foliation close to the unstable foliation of the map, endowed with a variant of the Wasserstein metric where mass is only allowed to be transported along the diffeomorphism's stable foliation. We show that this metric is indeed finite on that space, and use that the construction makes the diffeomorphism act as a contraction to deduce two corollaries. First, the SRB measure has exponential decay of correlation with respect to pairs of observable that are only asked to be H{ö}lder-continuous \emph{in the stable, respectively unstable direction}, but can be discontinuous overall. Then, we prove quantitative statistical stability: the map sending a $C^{1+α}$ Anosov diffeomorphism to its SRB measure is locally H{ö}lder-continuous (using the $C^1$ metric for diffeomorphisms and the usual Wasserstein metric for measures).

math.DS↗

arXiv2022

An invitation to rough dynamics: zipper maps

In the field of dynamical systems, it is not rare to meet irregular functions, which are typically H{ö}lder but not Lipschitz (e.g. the Weierstrass functions). Our goal is to scratch the surface of the following question: what happens if we consider irregular maps and iterate them? We introduce the family of "zipper maps", which are irregular in the above sense, and study some of their dynamical properties. For a large set of parameters, the corresponding zipper map admits horseshoe of all orders; as an immediate consequence, every order on $k\ell$ points can be realized by $k$ orbits of length $\ell$ of the map.These maps have infinite topological entropy, and we refine this statement by showing that they have positive metric mean dimension with respect to the Euclidean metric, as well as by introducing other notions of higher complexity.Finally, we prove that every interval map (thus including zipper maps) have vanishing absolute metric mean dimension, proving a small case of the conjecture that the absolute metric mean dimension coincides with the topological mean dimension.

math.DS↗

arXiv2022

An optimal transportation approach to the decay of correlations for non-uniformly expanding maps

We consider the transfer operators of non-uniformly expanding maps for potentials of various regularity, and show that a specific property of potentials ("flatness") implies a Ruelle-Perron-Frobenius Theorem and a decay of the transfer operator of the same speed than entailed by the constant potential. The method relies neither on Markov partitions nor on inducing, but on functional analysis and duality, through the simplest principles of optimal transportation. As an application, we notably show that for any map of the circle which is expanding outside an arbitrarily flat neutral point, the set of H{ö}lder potentials exhibiting a spectral gap is dense in the uniform topology. The method applies in a variety of situation, including Pomeau-Manneville maps with regular enough potentials, or uniformly expanding maps of low regularity with their natural potential; we also recover in a united fashion variants of several previous results.

math.CA↗

arXiv2022

Cycles in synchronous iterative voting: general robustness and examples in Approval Voting

We consider synchronous iterative voting, where voters are given the opportunity to strategically choose their ballots depending on the outcome deduced from the previous collective choices.We propose two settings for synchronous iterative voting, one of classical flavor with a discrete space of states, and a more general continuous-space setting extending the first one. We give a general robustness result for cycles not relying on a tie-breaking rule, showing that they persist under small enough perturbations of the behavior of voters. Then we give examples in Approval Voting of electorates applying simple, sincere and consistent heuristics (namely Laslier's Leader Rule or a modification of it) leading to cycles with bad outcomes, either not electing an existing Condorcet winner, or possibly electing a candidate ranked last by a majority of voters. Using the robustness result, it follows that those "bad cycles" persist even if only a (large enough) fraction of the electorate updates its choice of ballot at each iteration.We complete these results with examples in other voting methods, including ranking methods satisfying the Condorcet criterion; an in silico experimental study of the rarity of preference profiles exhibiting bad cycles; and an example exhibiting chaotic behavior.

cs.GT↗

arXiv2021

Nonlinear thermodynamical formalism

We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending investigation of the quadratic case by Leplaideur and Watbled. Under suitable conditions, we prove a variational principle for the nonlinear pressure and we characterize the nonlinear equilibrium measures and relate them to specific classical equilibrium measures. In this non-linear thermodynamical formalism, which can, e.g., model mean-field approximation of large systems, several kind of phase transitions appear, some of which cannot happen in the linear case. We use our correspondence between non-linear and linear equilibrium measures to further the understanding of phase transitions, { both in previously known cases (Curie-Weiss and Potts models) and in} new examples (metastable phase transition). Finally, we apply some of the ideas introduced to the classical thermodynamical formalism, proving that freezing phase transitions can occur over \emph{any} zero-entropy invariant compact subset of the phase space.

math.DS↗

arXiv2021

Optimal transportation and stationary measures for Iterated Function Systems

In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalized moment bounds from which tail estimates can be deduced, consider the convergence of the empirical measure of an associated Markov chain, and prove in many cases the Lipschitz continuity of the stationary measure when the system is perturbed, with as a consequence a "linear response formula" at almost every parameter of the perturbation.

math.CA↗

arXiv2021

Toy examples for effective concentration bounds

In this note we prove a spectral gap for various Markov chains on various functional spaces. While proving that a spectral gap exists is relatively common, explicit estimates seems somewhat rare.These estimates are then used to apply the concentration inequalities of "Effective limit theorems for Markov chains with a spectral gap" (most of the present material was part of Section 3 of that article, which has been reduced to its core in the published version).

math.DS↗

arXiv2019

A geometric study of Wasserstein spaces: isometric rigidity in negative curvature

Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X) when X is a simply connected, nonpositively curved metric spaces by considering its isometry group. When X is Euclidean, the second named author proved that this isometry group is larger than the isometry group of X. In contrast, we prove here a rigidity result: when X is negatively curved, any isometry of W2(X) comes from an isometry of X.

math.MG↗

arXiv2018

Effective limit theorems for Markov chains with a spectral gap

Applying quantitative perturbation theory for linear operators, we prove non-asymptotic limit theorems for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions X . The main results are concentration inequalities and Berry-Esseen bounds, obtained assuming neither reversibility nor `warm start' hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform X-ergodicity hypothesis, and when X consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the flexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.v2: Introduction rewritten, Section 3 applying the main results to examples improved (uniformly ergodic chains and Bernoulli convolutions have been notably added) . Main results and their proofs are unchanged.

math.PR↗

arXiv2018

Empirical measures: regularity is a counter-curse to dimensionality

We propose a "decomposition method" to prove non-asymptotic bound for the convergence of empirical measures in various dual norms. The main point is to show that if one measures convergence in duality with sufficiently regular observables, the convergence is much faster than for, say, merely Lipschitz observables. Actually, assuming $s$ derivatives with $s < d/2$ ($d$ the dimension) ensures an optimal rate of convergence of $1/\sqrt{n}$ ($n$ the number of samples). The method is flexible enough to apply to Markov chains which satisfy a geometric contraction hypothesis, assuming neither stationarity nor reversibility, with the same convergence speed up to a power of logarithm factor. Our results are stated as controls of the expected distance between the empirical measure and its limit, but we explain briefly how the classical method of bounded difference can be used to deduce concentration estimates.

math.PR↗

arXiv2017

Effective high-temperature estimates for intermittent maps

Using quantitative perturbation theory for linear operators, we prove spectral gap for transfer operators of various families of intermittent maps with almost constant potentials ("high-temperature" regime). Hölder and bounded p-variation potentials are treated, in each case under a suitable assumption on the map, but the method should apply more generally. It is notably proved that for any Pommeau-Manneville map, any potential with Lispchitz constant less than 0.0014 has a transfer operator acting on Lip([0, 1]) with a spectral gap; and that for any 2-to-1 unimodal map, any potential with total variation less than 0.0069 has a transfer operator acting on BV([0, 1]) with a spectral gap. We also prove under quite general hypotheses that the classical definition of spectral gap coincides with the formally stronger one used in (Giulietti et al. 2015), allowing all results there to be applied under the high temperature bounds proved here: analyticity of pressure and equilibrium states, central limit theorem, etc.

math.DS↗

arXiv2017

Mixed sectional-Ricci curvature obstructions on tori

We establish new obstruction results to the existence of Riemannian metrics on tori satisfying mixed bounds on both their sectional and Ricci curvatures. More precisely, from Lohkamp's theorem, every torus of dimension at least three admits Riemannian metrics with negative Ricci curvature. We show that the sectional curvature of these metrics cannot be bounded from above by an arbitrarily small positive constant. In particular, if the Ricci curvature of a Riemannian torus is negative, bounded away from zero, then there exist some planar directions in this torus where the sectional curvature is positive, bounded away from zero. All constants are explicit and depend only on the dimension of the torus.

math.DG↗

arXiv2017

The linear request problem

We propose a simple approach to a problem introduced by Galatolo and Pollicott, consisting in perturbing a dynamical system in order for its absolutely continuous invariant measure to change in a prescribed way. Instead of using transfer operators, we observe that restricting to an infinitesimal conjugacy already yields a solution. This allows us to work in any dimension and dispense from any dynamical hypothesis. In particular, we don't need to assume hyperbolicity to obtain a solution, although expansion moreover ensures the existence of an infinite-dimensional space of solutions.

math.DS↗

arXiv2017

The Cartan-Hadamard conjecture and The Little Prince

The generalized Cartan-Hadamard conjecture says that if $Ω$ is a domain with fixed volume in a complete, simply connected Riemannian $n$-manifold $M$ with sectional curvature $K \le κ\le 0$, then the boundary of $Ω$ has the least possible boundary volume when $Ω$ is a round $n$-ball with constant curvature $K=κ$. The case $n=2$ and $κ=0$ is an old result of Weil. We give a unified proof of this conjecture in dimensions $n=2$ and $n=4$ when $κ=0$, and a special case of the conjecture for $κ\textless{} 0$ and a version for $κ\textgreater{} 0$. Our argument uses a new interpretation, based on optical transport, optimal transport, and linear programming, of Croke's proof for $n=4$ and $κ=0$. The generalization to $n=4$ and $κ\ne 0$ is a new result. As Croke implicitly did, we relax the curvature condition $K \le κ$ to a weaker candle condition $Candle(κ)$ or $LCD(κ)$.We also find counterexamples to a naïve version of the Cartan-Hadamard conjecture: For every $\varepsilon \textgreater{} 0$, there is a Riemannian 3-ball $Ω$ with $(1-\varepsilon)$-pinched negative curvature, and with boundary volume bounded by a function of $\varepsilon$ and with arbitrarily large volume.We begin with a pointwise isoperimetric problem called "the problem of the Little Prince." Its proof becomes part of the more general method.

math.DG↗

arXiv2015

Geometry on the Utility Space

We study the geometrical properties of the utility space (the space of expected utilities over a finite set of options), which is commonly used to model the preferences of an agent in a situation of uncertainty. We focus on the case where the model is neutral with respect to the available options, i.e. treats them, a priori, as being symmetrical from one another. Specifically, we prove that the only Riemannian metric that respects the geometrical properties and the natural symmetries of the utility space is the round metric. This canonical metric allows to define a uniform probability over the utility space and to naturally generalize the Impartial Culture to a model with expected utilities.

cs.GT↗

arXiv2014

Curvatures and anisometry of maps

We prove various inequalities measuring how far from an isometry a local map from a manifold of high curvature to a manifold of low curvature must be. We consider the cases of volume-preserving, conformal and quasi-conformal maps. The proofs relate to a conjectural isoperimetric inequality for manifolds whose curvature is bounded above, and to a higher-dimensional generalization of the Schwarz-Ahlfors lemma.

math.DG↗

arXiv2014

A geometric study of Wasserstein spaces: ultrametrics

We study the geometry of the space of measures of a compact ultrametric space X, endowed with the L^p Wasserstein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely isometric to a convex subset of l^1. As a consequence, it is connected by 1/p-Hölder arcs, but any a-Hölder arc with a>1/p must be constant. This result is obtained via a reformulation of the distance between two measures which is very specific to the case when X is ultrametric; howeverthanks to the Mendel-Naor Ultrametric Skeleton it has consequences even when X is a general compact metric space. More precisely, we use it to estimate the size of Wasserstein spaces, measured by an analogue of Hausdorff dimension that is adapted to (some) infinite-dimensional spaces. The result we get generalizes greatly our previous estimate that needed a strong rectifiability assumption. The proof of this estimate involves a structural theorem of independent interest: every ultrametric space contains large co-Lipschitz images of \emph{regular} ultrametric spaces, i.e. spaces of the form {1,...,k}^N with a natural ultrametric. We are also lead to an example of independent interest: a space of positive lower Minkowski dimension, all of whose proper closed subsets have vanishing lower Minkowski dimension.

math.FA↗