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Jérôme Buzzi

Publications and source records attributed to Jérôme Buzzi.

16 recordsLinked to original sources
arXiv2026

Symbolic dynamics for non-uniformly hyperbolic flows

We construct symbolic dynamics for non-uniformly hyperbolic flows, in any dimension, possibly with fixed points. More precisely, for each $χ>0$, we code a set which has full measure for every $χ$-hyperbolic invariant probability measure that gives zero mass to the set of fixed points. As a main application, we prove that a three dimensional $C^\infty$ flow with positive topological entropy on a closed manifold has finitely many ergodic measures of maximal entropy. For flows in any dimension, we also provide applications to the number of periodic orbits and to the Bernoulli property for equilibrium states of Hölder continuous potentials. The main technical result of this paper is a new method for handling singularities of vector fields by modifying the Riemannian metric. This technique is analogous to a blowup, which allows many results for nonsingular vector fields to be directly applied to vector fields with singularities.

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arXiv2026

Margulis Measures on Expanding Foliations: Construction and Rigidity

Given a diffeomorphism preserving a one-dimensional expanding foliation $\mathcal F$ with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property. We then prove that for any measure of maximal $u$-entropy, its conditional measures on each leaf must be equivalent to the reference measures. When the measure of maximal $u$-entropy is a Gibbs $\mathcal F$-state (i.e., when the reference measures are equivalent to the leafwise Lebesgue measure), we prove that the log-Jacobian of $f$ must be cohomologous to a constant via a measurable function. We provide several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature.

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arXiv2026

Chaos on surfaces and beyond: a new notion of dynamical hyperbolicity

We present some developments in the study of chaotic dynamics following the solution of a conjecture of Newhouse on the measures maximizing the entropy of smooth surface diffeomorphisms. We focus on \emph{strong positive recurrence}, a generalization of the classical Anosov-Smale theory of uniform hyperbolicity introduced in a joint work with Sylvain Crovisier and Omri Sarig. This new property is general enough to be satisfied by all smooth surface diffeomorphisms with positive entropy, yet it still ensures many quantitative properties such as exponential mixing or limit theorems for regular functions. We also present some open problems, including its abundance (or not) in higher dimensions.

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arXiv2026

Discontinuity of Lyapunov exponents vs Entropy for smooth surface diffeomorphisms

Lyapunov exponents are fundamental invariants in smooth ergodic theory describing the asymptotic infinitesimal behavior along typical orbits. This text aims to explain how and why to control Lyapunov exponents using entropy for smooth surface diffeomorphisms. It fits into the framework of our recent joint works with Sylvain CROVISIER and Omri SARIG. We will focus especially on the continuity property of exponents for measures near the maximal entropy measure, by presenting a simplified version of the original argument. Our exposition is geared towards advanced students and researchers in dynamics that are not necessarily familiar with smooth ergodic theory.

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arXiv2025

Continuity properties of ergodic measures of maximal entropy for $C^r$ surface diffeomorphisms

Let $f$ be a $C^r$ surface diffeomorphism with large entropy (more precisely, $h_{\rm top}(f)>λ_{\min}(f)/{r}$). Then the number of ergodic measures of maximal entropy is upper semicontinuous at $f$. This generalizes the $C^\infty$ case studied in \cite{BCS22}, answering Question 1.9 there. Moreover, the number of such measures is locally constant if and only if every ergodic measure of maximal entropy of $f$ admits an ergodic continuation under small perturbations. In this case, the accumulation points of ergodic measures of maximal entropy are themselves ergodic. These facts are new, even in the $C^\infty$ case.

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arXiv2025

Strong positive recurrence and exponential mixing for diffeomorphisms

We introduce the strong positive recurrence (SPR) property for diffeomorphisms on closed manifolds with arbitrary dimension, and show that it has many consequences and holds in many cases. SPR diffeomorphisms can be coded by countable state Markov shifts whose transition matrices act with a spectral gap on a large Banach space, and this implies exponential decay of correlations, almost sure invariance principle, large deviations, among other properties of the ergodic measures of maximal entropy. Any $C^\infty$ smooth surface diffeomorphism with positive entropy is SPR, and there are many other examples with lesser regularity, or in higher dimension.

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arXiv2023

Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows

We construct symbolic dynamics for three dimensional flows with positive speed. More precisely, for each $χ>0$, we code a set of full measure for every invariant probability measure which is $χ$-hyperbolic. These include all ergodic measures with entropy bigger than $χ$ as well as all hyperbolic periodic orbits of saddle-type with Lyapunov exponent outside of $[-χ,χ]$. This contrasts with a previous work of Lima & Sarig which built a coding associated to a given invariant probability measure. As an application, we code homoclinic classes of measures by suspensions of irreducible countable Markov shifts.

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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.

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arXiv2021

Continuity properties of Lyapunov exponents for surface diffeomorphisms

We study the entropy and Lyapunov exponents of invariant measures $μ$ for smooth surface diffeomorphisms $f$, as functions of $(f,μ)$. The main result is an inequality relating the discontinuities of these functions. One consequence is that for a $C^\infty$ surface diffeomorphisms, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy.

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arXiv2021

Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature

We study the one parameter family of potential functions $qφ^u$ associated with the geometric potential $φ^u$ for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For $q<1$ it is known that there is a unique equilibrium state associated with $qφ^u$, and it has full support. For $q > 1$ it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value $q=1$ and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure, or measures supported on the singular set. In particular, when~$q = 1$, there is a unique ergodic equilibrium state that gives positive measure to the regular set.

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arXiv2020

A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows

We show that time-one maps of transitive Anosov flows of compact manifolds are accumulated by diffeomorphisms robustly satisfying the following dichotomy: either all of the measures of maximal entropy are non-hyperbolic, or there are exactly two ergodic measures of maximal entropy, one with a positive central exponent and the other with a negative central exponent. We establish this dichotomy for certain partially hyperbolic diffeomorphisms isotopic to the identity whenever both of their strong foliations are minimal. Our proof builds on the approach developed by Margulis for Anosov flows where he constructs suitable families of measures on the dynamical foliations.

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arXiv2019

Measures of maximal entropy for surface diffeomorphisms

We show that $C^\infty$ surface diffeomorphisms with positive topological entropy have at most finitely many ergodic measures of maximal entropy in general, and at most one in the topologically transitive case. This answers a question of Newhouse, who proved that such measures always exist. To do this we generalize Smale's spectral decomposition theorem to non-uniformly hyperbolic surface diffeomorphisms, we introduce homoclinic classes of measures, and we study their properties using codings by irreducible countable state Markov shifts.

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arXiv2019

Large entropy implies existence of a maximal entropy measure for interval maps

We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map $f$, using some non-uniform hyperbolicity to compensate for a lack of smoothness of $f$. More precisely, if the topological entropy of a $C^1$ interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any $C^r$ interval map $f$ such that $h_{\rm top}(f)>2\log\|f'\|_{\infty}/r$ possesses measures with maximal entropy.

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arXiv2015

The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors

Extending work of Hochman, we study the almost-Borel structure, i.e., the nonatomic invariant probability measures, of symbolic systems and surface diffeomorphisms. We first classify Markov shifts and characterize them as strictly universal with respect to a natural family of classes of Borel systems. We then study their continuous factors showing that a low entropy part is almost-Borel isomorphic to a Markov shift but that the remaining part is much more diverse, even for finite-to-one factors. However, we exhibit a new condition which we call "Bowen type" which gives complete control of those factors. This last result applies to and was motivated by the symbolic covers of Sarig. We find complete numeric invariants for Borel isomorphism of $C^{1+}$ surface diffeomorphisms modulo zero entropy measures; for those admitting a totally ergodic measure of positive (not necessarily maximal) entropy, we get a classification up to almost-Borel isomorphism.

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