February 4th, 2021


Zoom coordinates:
Zoom meeting number: 998 2859 4541
Password hint: a numerical invariant which measures complexity of a dynamical system (one word, lower case)


"A Hyperbolic Day Online" is a one-day meeting focused on topics around hyperbolic dynamics. The conference will use zoom. All are welcome to participate.
The invitations to zoom talks will be distributed by email to registered participants. We encourage the participants to register (which takes less than one minute) in order to receive all relevant communications by email. However we will also post the zoom meeting number and a hint for the password on this page in the morning on February 4th.
Available slides will be posted on this page after the conference concludes, but we won't make video recordings of the talks.

Organizers: Andrey Gogolev (Ohio State), Rafael Potrie (Universidad de la Republica)

Registration

Format

The meeting will feature 7 short talks and a plenty of time for follow-up discussions and exchange of ideas. The talks will be 20 minutes long followed by a 20 minute period of questions and discussion with the audience. Everybody are welcome to participate in the discussion period which will be facilitated by the talk chair. We ask all participants to use their real names on zoom and turn on cameras when asking questions.

Invited speakers

Anna Florio (Université Paris Dauphine) Smooth conjugacy classes of 3D Axiom A flows
Katrin Gelfert (Universidade Federal do Rio de Janeiro) Expansive flow-models for geodesic flows
Tali Pinsky (Technion) A three manifold carrying infinitely many Anosov flows
Mauricio Poletti (Universidade Federal do Ceará) Partially hyperbolic diffeomorphisms with zero center exponent
Mario Shannon (Universite Aix-Marseille) Hyperbolic models of transitive topological Anosov flows
Yi Shi (Peking University) Spectrum rigidity and integrability for codimension-one Anosov diffeomorphisms
Khadim War (IMPA) Closed geodesics on surfaces without conjugate points

Schedule for February 4th

Speaker Chair Time
Tali Pinsky Christian Bonatti 9:00-9:40am (New York) = 11:00-11:40am (Rio de Janeiro) = 3:00-3:40pm (Paris)
Yi Shi Amie Wilkinson 9:50-10:30am (New York) = 11:50-12:30pm (Rio de Janeiro) = 3:50-4:30pm (Paris)
Mauricio Poletti Enrique Pujals 10:40-11:20am (New York) = 12:40-1:20pm (Rio de Janeiro) = 4:40-5:20pm (Paris)
Lunch break
   
Anna Florio Federico Rodriguez Hertz 1:00-1:40pm (New York) = 3:00-3:40pm (Rio de Janeiro) = 7:00-7:40pm (Paris)
Mario Shannon Sergio Fenley 1:50-2:30pm (New York) = 3:50-4:30pm (Rio de Janeiro) = 7:50-8:30pm (Paris)
Coffee break
   
Khadim War Keith Burns 3:00-3:40pm (New York) = 5:00-5:40pm (Rio de Janeiro) = 9:00-9:40pm (Paris)
Katrin Gelfert Bruno Santiago 3:50-4:30pm (New York) = 5:50-6:30pm (Rio de Janeiro) = 9:50-10:30pm (Paris)


Abstracts

Anna Florio
Smooth conjugacy classes of 3D Axiom A contact flows
In a joint work with Martin Leguil, we consider an orbit equivalence between the restriction of two 3-dimensional Axiom A contact flows on some basic set. If the orbit equivalence is iso-length-spectral, then we show that the dynamics on the basic sets are conjugated through a homeomorphism, as regular as the flows in Whitney sense, which also preserves the contact structures.

Katrin Gelfert
Expansive flow-models for geodesic flows
We study a class of geodesic flows of compact surfaces (of higher genus and either without focal points or without conjugate points and continuous Green bundles), aiming to understand their thermodynamical properties. We show that each such flow is semi-conjugate to a topological Anosov flow on some 3-dimensional manifold by some map which preserves time parametrization. This can successfully be used to investigate measures of maximal entropy and equilibrium states. This talk is about joint work with Rafael Ruggiero and Dominik Kwietniak.

Tali Pinsky
A three manifold carrying infinitely many Anosov flows
The geodesic flow on the modular surface is an Anosov flow on the complement of a trefoil knot in S^3. In the talk I will describe how to use the infinitely many self covers of the trefoil complement in order to lift the geodesic flow, and obtain infinitely many Anosov flows on the trefoil complement. For all of these flows, the boundary of the trefoil complement is a Birkhoff torus with two tangent orbits. Conjecturally, one can glue together two copies of the trefoil complement, and obtain a single closed manifold with infinitely many smooth flows. This is joint work with Adam Clay.

Mauricio Polletti
Partially hyperbolic diffeomorphisms with zero center exponent
Ledrappier proved that the invariant measures of linear cocycles having zero Lyapunov exponents have certain extra invariance. This was generalized by Avila and Viana for smooth cocycles, in particular they proved that the invariant measures for partially hyperbolic skew products have a disintegration invariant by holonomies, this is known as "invariance principle". This has several applications, such as obtaining genericity of non-uniformly hyperbolic systems, finding physical measures, and classifying the measures of maximal entropy.
In this presentation we will generalize the invariance principle to partially hyperbolic non-skew products (without compact center leaves) which allows us to extend several of the previous applications to more general partially hyperbolic ones. In particular we will give an application to classify measures of maximal entropy of the perturbation of the time one map of Anosov flows. This is a joint work with Sylvain Crovisier.

Mario Shannon
Hyperbolic models of transitive topological Anosov flows
A topological Anosov flow on a closed 3-manifold is a non-singular flow that resembles very much a smooth Anosov flow: It is expansive and satisfies the (global) shadowing property. Moreover, it preserves a pair of transverse stable/unstable foliations that intersect along the orbits of the flow. The main difference with a (smooth) Anosov flow is the lack of a global uniformly hyperbolic structure. We investigate the question of weather or not every topological Anosov flow in a 3-manifold is, actually, orbit equivalent with some smooth Anosov flow. Apart from its own theoretical interest, this question appears related with some techniques for the construction of Anosov flows on 3-manifolds, notably with the so called Fried surgery. Our work consists in show that, under the hypothesis of transitivity, every topological Anosov flow is orbitally equivalent with a smooth Anosov flow. In this talk we are going to present the main ideas and difficulties behind the construction of these smooth hyperbolic models associated with transitive topological Anosov flows. As well, we will give a flavour of how this study can be reduced to pseudo-Anosov dynamics on surfaces with boundary.

Yi Shi
Spectrum rigidity and integrability for codimension-one Anosov diffeomorphisms
Let A be a linear hyperbolic automorphism with one-dimensional unstable bundle. We show that for every f close to A, if the strong stable bundle and the unstable bundle of f are jointly integrable, then every periodic point of f has the same Lyapunov exponents with A along all weak stable bundles. This is a joint work with Andrey Gogolev.

Khadim War
Closed geodesics on surfaces without conjugate points
We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points. This is based on a join work with Vaughn Climenhaga and Gerhard Knieper.