Jérôme Buzzi
YOU?
Author Swipe
View article: Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows
Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows Open
79 pages, 5 figures
View article: Strong positive recurrence and exponential mixing for diffeomorphisms
Strong positive recurrence and exponential mixing for diffeomorphisms Open
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 stat…
View article: Nonlinear thermodynamical formalism
Nonlinear thermodynamical formalism Open
We define a nonlinear thermodynamical formalism which translates into dynamical system theory the statistical mechanics of generalized mean-field models, extending the investigation of the quadratic case in one or more potentials by Leplai…
View article: Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows
Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows Open
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 measur…
View article: Open questions in descriptive set theory and dynamical systems
Open questions in descriptive set theory and dynamical systems Open
This file is composed of questions that emerged or were of interest during the workshop "Interactions between Descriptive Set Theory and Smooth Dynamics" that took place in Banff, Canada on 2022.
View article: Another proof of Burguet's existence theorem for SRB measures of $C^\infty$ surface diffeomorphisms
Another proof of Burguet's existence theorem for SRB measures of $C^\infty$ surface diffeomorphisms Open
Recently, Burguet proved a strong form of Viana's conjecture on physical measures, in the special case of $C^\infty$ surface diffeomorphisms. We give another proof, based on our analysis of entropy and Lyapunov exponents in [BCS].
View article: Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature
Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature Open
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 equilibr…
View article: Continuity properties of Lyapunov exponents for surface diffeomorphisms
Continuity properties of Lyapunov exponents for surface diffeomorphisms Open
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 tha…
View article: Phase transitions for the geodesic flow of a rank one surface with\n nonpositive curvature
Phase transitions for the geodesic flow of a rank one surface with\n nonpositive curvature Open
We study the one parameter family of potential functions $q\\varphi^u$\nassociated with the geometric potential $\\varphi^u$ for the geodesic flow of a\ncompact rank 1 surface of nonpositive curvature. For $q<1$ it is known that\nthere is …
View article: Nonlinear thermodynamical formalism
Nonlinear thermodynamical formalism Open
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 suitab…
View article: The degree of Bowen factors and injective codings of diffeomorphisms
The degree of Bowen factors and injective codings of diffeomorphisms Open
We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce Hölder-continuous conjugacies on large sets. We deduce this from their Bowen property. This notion, introduced in a joint…
View article: A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows
A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows Open
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 exact…
View article: Large entropy implies existence of a maximal entropy measure for interval maps
Large entropy implies existence of a maximal entropy measure for interval maps Open
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 e…
View article: Bowen factors, their degree, and codings of surface diffeomorphisms
Bowen factors, their degree, and codings of surface diffeomorphisms Open
We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce H\older-continuous conjugacies on large sets, sometimes preserving transitivity. We deduce this from their Bowen property…
View article: The entropy of 𝐶¹-diffeomorphisms without a dominated splitting
The entropy of 𝐶¹-diffeomorphisms without a dominated splitting Open
A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local -perturbations. Our main theorem shows that when one works in the -topology, the entropy of suc…
View article: The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors
The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors Open
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 universa…
View article: Entropy of $C^1$ diffeomorphisms without a dominated splitting
Entropy of $C^1$ diffeomorphisms without a dominated splitting Open
A classical construction due to Newhouse creates horseshoes from hyperbolic\nperiodic orbits with large period and weak domination through local\n$C^1$-perturbations. Our main theorem shows that, when one works in the $C^1$\ntopology, the …
View article: The almost Borel structure of diffeomorphisms with some hyperbolicity
The almost Borel structure of diffeomorphisms with some hyperbolicity Open
These lecture notes focus on a recent result of Mike Hochman: an arbitrary\nstandard Borel system can be embedded into a mixing Markov with equal entropy,\nrespecting all invariant probability measures, with two exceptions: those\ncarried …