arXiv subjects

Dominique Malicet

Publications and source records attributed to Dominique Malicet.

At least 19 recordsLinked to original sources

arXiv2026

(co)Quasi-irreducible and (co)expanding random maps

We study quasi-irreducibility, expansion, and their cotangent duals for random $C^1$ (local) diffeomorphisms on compact invariant sets of a Riemannian manifold. Quasi-irreducibility is defined through stationary lifts to the projective tangent bundle: every lift must realize the Furstenberg--Kifer formula for the top Lyapunov exponent. We prove that, for ergodic stationary measures, this is equivalent to the absence of equators, namely non-random invariant subbundles on which the top exponent drops. Under simplicity of the first Lyapunov exponent for every stationary measure, quasi-irreducibility is also equivalent to vertical mostly contraction, contraction on average, a vertical spectral gap, and uniqueness of stationary projective lifts. We then apply these criteria to continuity of the top Lyapunov exponent and to expansion on average. In particular, expansion is characterized by positivity of the top Lyapunov exponent on all non-random invariant subbundles; under quasi-irreducibility, it is equivalent to positivity of the top Lyapunov exponent for every stationary measure. Dual statements hold for coquasi-irreducibility, coequators, coexpansion, and continuity of the bottom Lyapunov exponent. We also describe the interplay between expansion and coexpansion and extend the formalism to Grassmannian bundles, obtaining higher-dimensional versions controlling intermediate sums of Lyapunov exponents.

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arXiv2025

Finitude of physical measures for Markovian random maps

We study the finiteness of physical measures for skew-product transformations $F$ associated with discrete-time random dynamical systems driven by ergodic Markov chains. We develop a framework, using an independent and identically distributed (i.i.d.) representation of the Markov process, that facilitates transferring results from the well-studied Bernoulli (i.i.d.) setting to the Markovian context. Specifically, we establish conditions for the existence of finitely many ergodic, $F$-invariant measures, absolutely continuous with respect to a reference measure, such that their statistical basins of attraction for measurable bounded observables cover the phase space almost everywhere. Furthermore, we investigate a weaker notion, which demands finitely many physical measures (not necessarily absolutely continuous) whose weak$^*$ basins of attraction cover the phase space almost everywhere. We show that for random maps on compact metric spaces driven by Markov chains on finite state spaces, this property holds if the system is mostly contracting, i.e., if all the Markovian invariant measures have negative maximal Lyapunov exponents. This result is applied to random $C^1$ diffeomorphisms of the circle and the interval under conditions based on the absence of invariant probability measures or finite invariant sets, respectively. We also connect our result to the quasi-compactness of the Koopman operator on the space of Hölder continuous functions.

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arXiv2025

Random Dynamical Systems on the circle without a finite orbit

In this paper, we study Random Dynamical Systems (RDSs) of homeomorphisms on the circle without a finite orbit. We characterize the topological dynamics of the associated semigroup by identifying the existence of invariant sets which are finite unions of intervals. We describe the accumulation points of the average orbit of the transfer operator. For each ergodic stationary measure, we demonstrate interesting properties of its weight function on the circle. Relationships between the minimal sets of an RDS and its inverse RDS are also established.

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arXiv2024

Limit distributions for polynomials with independent and identically distributed entries

We characterize the limiting distributions of random variables of the form $P_n\left( (X_i)_{i \ge 1} \right)$, where: (i) $(P_n)_{n \ge 1}$ is a sequence of multivariate polynomials, each potentially involving countably many variables; (ii) there exists a constant $D \ge 1$ such that for all $n \ge 1$, the degree of $P_n$ is bounded above by $D$; (iii) $(X_i)_{i \ge 1}$ is a sequence of independent and identically distributed random variables, each with zero mean, unit variance, and finite moments of all orders. More specifically, we prove that the limiting distributions of these random variables can always be represented as the law of $P_\infty\left( (X_i, G_i)_{i \ge 1} \right)$, where $P_\infty$ is a polynomial of degree at most $D$ (potentially involving countably many variables), and $(G_i)_{i \ge 1}$ is a sequence of independent standard Gaussian random variables, which is independent of $(X_i)_{i \ge 1}$. We solve this problem in full generality, addressing both Gaussian and non-Gaussian inputs, and with no extra assumption on the coefficients of the polynomials. In the Gaussian case, our proof builds upon several original tools of independent interest, including a new criterion for central convergence based on the concept of maximal directional influence. Beyond asymptotic normality, this novel notion also enables us to derive quantitative bounds on the degree of the polynomial representing the limiting law. We further develop techniques regarding asymptotic independence and dimensional reduction. To conclude for polynomials with non-Gaussian inputs, we combine our findings in the Gaussian case with invariance principles.

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arXiv2024

Mostly contracting random maps

We study the long-term behavior of independent random iterations of Lipschitz transformations on a compact metric space. Such a random map is said to be mostly contracting if all Lyapunov exponents associated with stationary measures are negative. This requires introducing the notion of (maximal) Lyapunov exponent in this general context of Lipschitz transformations on compact metric spaces. We show that this class is open with respect to the appropriate topology and satisfies the strong law of large numbers for non-uniquely ergodic systems, the limit theorem for the law of random iterations, Palis' global conjecture, and that the associated annealed Koopman operator is quasi-compact. This implies many statistical properties such as central limit theorems, large deviations, statistical stability, and the continuity and H\"older continuity of Lyapunov exponents. Examples from this class of random maps include random products of circle diffeomorphisms, interval diffeomorphisms onto their images, and diffeomorphisms of a Cantor set on a line, all considered under the assumption of no common invariant measure. This class also includes projective actions of locally constant linear cocycles under the assumptions of simplicity of the top Lyapunov exponent and a suitable irreducibility condition. Finally, it encompasses the classical theory of finite-state Markov chains, viewed as random walks on a finite set. One of the main tools to prove the above results is the generalization of Kingman's subadditive ergodic theorem and the uniform Kingman's subadditive ergodic theorem for general Markov operators. Another key ingredient is the exponential local contraction theorem, which can be viewed as a non-smooth metric analogue of the contraction mechanism behind stable manifold theory. These results are of independent interest, as they may have broad applications in other contexts.

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arXiv2024

Random actions of homeomorphisms of Cantor sets embedded in a line and Tits alternative

In 2000, Margulis proved that any group of homeomorphisms of the circle either preserves a probabilty measure on the circle or contains a free subgroup in two generators, which is reminiscent of the Tits alternatve for linear groups. In this article, we prove an analogous statement for groups of locally monotonic homeomorphisms of a compact subset of R.

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arXiv2024

Regularity of laws via Dirichlet forms -- Application to quadratic forms in independent and identically distributed random variables

We study the regularity of the law of a quadratic form $Q(X,X)$, evaluated in a sequence $X = (X_{i})$ of independent and identically distributed random variables, when $X_{1}$ can be expressed as a sufficiently smooth function of a Gaussian field. This setting encompasses a large class of important and frequently used distributions, such as, among others, Gaussian, Beta, for instance uniform, Gamma distributions, or else any polynomial transform of them. Let us present an emblematic application. Take $X = (X_{i})$ a sequence of independent and identically distributed centered random variables, with unit variance, following such distribution. Consider also $(Q_{n})$ a sequence of quadratic forms, with associated symmetric Hilbert--Schmidt operators $(\mathsf{A}^{(n)})$. Assume that $\operatorname{Tr}[ (\mathsf{A}^{(n)})^{2} ] = 1/2$, $\mathsf{A}^{(n)}_{ii} =0$, and the spectral radius of $\mathsf{A}^{(n)}$ tends to $0$. Then, $(Q_{n}(X))$ converges in a strong sense to the standard Gaussian distribution. Namely, all derivatives of the densities, which are well-defined for $n$ sufficiently large, converge uniformly on $\mathbb{R}$ to the corresponding derivatives of the standard Gaussian density. While classical methods, from Malliavin calculus or $Γ$-calculus, generally consist in bounding negative moments of the so-called \emph{carré du champ} operator $Γ(Q(X),Q(X))$, we provide a new paradigm through a second-order criterion involving the eigenvalues of a Hessian-type matrix related to $Q(X)$. This Hessian is built by iterating twice a tailor-made gradient, the \emph{sharp operator} $\sharp$, obtained via a Gaussian representation of the carré du champ. We believe that this method, recently developed by the authors in the current paper and in their companion paper [AoP 52 n°3 (2024)] , is of independent interest and could prove useful in other settings.

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arXiv2024

Sharp total variation rates of convergence for fluctuations of linear statistics of $β$-ensembles

In this article, we revisit the question of fluctuations of linear statistics of beta ensembles in the single cut and non-critical regime for general potentials $V$ under mild regularity and growth assumptions. Our main objective is to establish sharp quantitative Central Limit Theorems (CLT) for strong distances, such as the total variation distance, which to the best of our knowledge, is new for general potentials, even qualitatively. Namely, setting $μ_V$ the equilibrium measure, for a test function $ξ\in \mathscr{C}^{14}$, we establish the convergence in total variation of $X_n=\sum_{i=1}^n ξ(λ_i)-n\langle ξ,μ_V\rangle$ to an explicit Gaussian variable at the sharp speed $1/n$. Under the same assumptions, we also establish multivariate CLTs for vectors of linear statistics in $p-$Wasserstein distances for any $p\ge 1$, with the optimal rate $1/n$, a result which already in dimension one sharpens the speed of convergence established in the recent contribution [26] as well as the required regularity on the test functions. A second objective of this paper, in a more qualitative direction, is to establish the so-called super-convergence of linear statistics, that is to say the convergence of all derivatives of the densities of $X_n$ uniformly on $\mathbb{R}$, provided that $ξ\in\mathscr{C}^\infty(\mathbb{R})$ and is not too degenerated in some sense.

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arXiv2024

Superconvergence phenomenon in Wiener chaoses

We establish an unexpected phenomenon of strong regularization along normal convergence on Wiener chaoses. For every sequence of chaotic random variables, convergence in law to the Gaussian distribution is upgraded to superconvergence: the regularity of the densities increases along the convergence, and all the derivatives converges uniformly. Our findings strengthen known results regarding modes of convergence for normal approximation on Wiener chaoses. Without additional assumptions, convergence in total variation is established by Nourdin & Peccati, and later on amplified to convergence in relative entropy by Nourdin, Peccati & Swan. Our result is then extended to the multivariate setting, and for polynomial mappings of a Gaussian field provided the projection on the Wiener chaos of maximal degree admits a non-degenerate Gaussian limit. While our findings apply to any context involving polynomials of a Gaussian field, we emphasize applications regarding: improved Carbery-Wright estimates near Gaussianity; normal convergence in entropy and in Fisher information; superconvergence for the spectral moments of GOE; moments bounds for the inverse of strongly correlated Wishart-type matrices; superconvergence in the Breuer-Major Theorem. Our proofs leverage Malliavin's historical idea to establish smoothness of the density via the existence of negative moments of the Malliavin gradient, and we develop a new paradigm to study this problem. We relate the existence of negative moments to spectral quantities associated with the Malliavin Hessian. This link relies on an adequate choice of the Malliavin gradient, which provides a novel decoupling procedure of independent interest. Previous attempts to establish convergence beyond entropy have imposed restrictive assumptions ensuring finiteness of negative moments for the Malliavin derivatives. Our analysis renders these assumptions superfluous.

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arXiv2023

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, I: Construction

Following the recent advances in the study of groups of circle diffeomorphisms, we describe an efficient way of classifying the topological dynamics of locally discrete, finitely generated, virtually free subgroups of the group $\mathsf{Diff}^ω_+(\mathbb S^1)$ of orientation preserving real-analytic circle diffeomorphisms, which include all subgroups of $\mathsf{Diff}^ω_+(\mathbb S^1)$ acting with an invariant Cantor set. An important tool that we develop, of independent interest, is the extension of classical ping-pong lemma to actions of fundamental groups of graphs of groups. Our main motivation is an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403--453] from foliation theory, which we solve in this restricted but significant setting: this and other consequences of the classification will be treated in more detail in a companion work.

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arXiv2023

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications

In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403-453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.

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arXiv2023

Groups of smooth diffeomorphisms of Cantor sets embedded in a line

Let K be a Cantor set embedded in the real line R. Following Funar and Neretin, we define the diffeomorphism group of K as the group of homeomorphisms of K which locally look like a diffeomorphism between two intervals of R. Higman-Thompson's groups Vn appear as subgroups of such groups. In this article, we prove some properties of this group. First, we study the Burnside problem in this group and we prove that any finitely generated subgroup consisting of finite order elements is finite. This property was already proved by Rover in the case of the groups Vn. We also prove that any finitely generated subgroup H without free subsemigroup on two generators is virtually abelian. The corresponding result for the groups Vn was unknown to our knowledge. As a consequence, those groups do not contain nilpotent groups which are not virtually abelian.

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arXiv2023

Almost sure behavior of the critical points of random polynomials

Let $(Z_k)_{k\geq 1}$ be a sequence of independent and identically distributed complex random variables with common distribution $μ$ and let $P_n(X):=\prod_{k=1}^n (X-Z_k)$ the associated random polynomial in $\mathbb C[X]$. In [Kab15], the author established the conjecture stated by Pemantle and Rivin in [PR13] that the empirical measure $ν_n$ associated with the critical points of $P_n$ converges weakly in probability to the base measure $μ$. In this note, we establish that the convergence in fact holds in the almost sure sense. Our result positively answers a question raised by Z. Kabluchko and formalized as a conjecture in the recent paper [MV22].

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arXiv2022

A short proof of the strong three dimensional Gaussian product inequality

We prove the strong form of the Gaussian product conjecture in dimension three. Our purely analytical proof simplifies previously known proofs based on combinatorial methods or computer-assisted methods, and allows us to solve the case of any triple of even positive integers which remained open so far.

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arXiv2021

Extremal exponents of random products of conservative diffeomorphisms

We show that for a $C^1$-open and $C^{r}$-dense subset of the set of ergodic iterated function systems of conservative diffeomorphisms of a finite-volume manifold of dimension $d\geq 2$, the extremal Lyapunov exponents do not vanish. In particular, the set of non-uniform hyperbolic systems contains a $C^1$-open and $C^r$-dense subset of ergodic random products of i.i.d. conservative surface diffeomorphisms.

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arXiv2020

Lyapunov exponent of random dynamical systems on the circle

We consider products of a i.i.d. sequence in a set $\{f_1,\ldots,f_m\}$ of preserving orientation diffeomorphisms of the circle. we can naturally associate a Lyapunov exponent $λ$. Under few assumptions, it is known that $λ\leq 0$ and that the equality holds if and only if $f_1,\ldots,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots,f_m$ are $C^k$ perturbations of rotations with rotation numbers $ρ(f_1),\ldots,ρ(f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser: we give a precise estimate on $λ$ (Taylor expansion) and we prove that there exists a diffeomorphism $g$ and rotations $r_i$ such that $\mbox{dist}(gf_ig^{-1},r_i)\ll |λ|^{\frac{1}{2}}$ for $i=1,\ldots m$. We also state analog results for random products of matrices $2\times 2$, without diophantine condition.

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arXiv2019

Entropy, Lyapunov exponents, and rigidity of group actions

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teresópolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems 16, 1996] and recent the work of the main author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [arXiv:1608.04995; arXiv:1610.09997]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures.

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arXiv2018

Groups with infinitely many ends acting analytically on the circle

This article takes the inspiration from two milestones in the study of non minimal actions of groups on the circle: Duminy's theorem about the number of ends of semi-exceptional leaves and Ghys' freeness result in analytic regularity. Our first result concerns groups of analytic diffeomorphisms with infinitely many ends: if the action is non expanding, then the group is virtually free. The second result is a Duminy's theorem for minimal codimension one foliations: either non expandable leaves have infinitely many ends, or the holonomy pseudogroup preserves a projective structure.

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