Scientific Calendar Event

 Description https://www.youtube.com/playlist?list=PLLq_gUfXAnkm6iDpTP4qzbGbXJPyxt1nb https://www.youtube.com/playlist?list=PLLq_gUfXAnkk84vDzT4pPLIXgifiLkS3g https://www.youtube.com/playlist?list=PLLq_gUfXAnkmuZtzDtmz6Sp1dJJt_dEpY In the 1970s, Sinai, Ruelle, and Bowen, developed groundbreaking new ideas and techniques which made it possible to apply the powerful results of Ergodic Theory to concrete, and sometimes quite explicit, differentiable dynamical systems. In particular they showed that smooth Uniformly Hyperbolic systems admit Markov Partitions, from which one can obtain a Symbolic Coding with a finite number of symbols. This Symbolic Coding makes it possible to apply methods from statistical mechanics to describe the statistical properties of the system through the construction of a particular class of invariant measures which are now called Sinai-Ruelle-Bowen (SRB) measures. Over the last 20 years there has been a huge progress in extending the results of Sinai, Ruelle, and Bowen, to the much larger class of more general (Nonuniformly) Hyperbolic systems, including systems with discontinuities/singularities. The geometry of these systems is much more complicated and one cannot expect to be able to code them with a symbolic dynamics with a finite number of symbols, making them much more challenging to study. Two inter-related but distinct approaches have emerged. At the end of the 1990s, Lai-Sang Young introduced a construction which is now generally refereed to as a Young Tower, based on constructing an induced uniformly hyperbolic system within the given system. More recently, around 2013, Sarig generalised the original Sinai-Ruelle-Bowen approach to construct infinite Markov Partitions. Both approaches have proved quite powerful and have been used to construct SRB measures and to study their statistical properties in a number of classes of dynamical systems of great interest. On the other hand, both approaches are also technically non-trivial and as a consequence, notwithstanding the applications of both to similar systems and their inevitable underlying connections, most researchers have developed an expertise in either one or the other. The main purpose of this event is to bring together experts in both areas in order to create opportunities to understand better the similarities and differences between them, and the advantages and disadvantages of the two approaches. The entire event will be held online with 2 mini courses by José Ferreira Alves on Young Towers and Yuri Lima on Markov Partitions. These mini-courses will be introductory and require only some familiarity with Uniformly Hyperbolic Dynamics and will be spread out over a period of 2 to 3 weeks, and be accompanied by some additional tutorial sessions, in order to give participants time to actually study the material and consolidate their knowledge. The mini courses will then be followed by a week of research level seminars describing recent results on these topics. Mini-Course Lecturers: José Ferreira Alves, U. of Porto, Portugal Yuri Lima, UFC, Brazil Speakers: Jerome Buzzi (Paris) Jianyu Chen (Soochow U.) Sylvain Crovisier (Paris) Peyman Eslami (Rome) Carlos Matheus (Paris) Snir Ben Ovadia (Weizman) Yakov Pesin (Penn State) Vilton Pinheiro (UFBA) Omri Sarig (Weizman) Agnieska Zelerowicz (U. Maryland)

### Material

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• Monday, 15 November 2021
• 09:00 - 17:00 No lectures on Monday 15 Nov 2021
• Tuesday, 16 November 2021
• 14:00 - 16:30 Symbolic dynamics for nonuniformly hyperbolic systems  Convener: Yuri Lima (UFC, Brazil)
• Wednesday, 17 November 2021
• 09:00 - 17:00 No lectures on Wed 17 Nov 2021
• Thursday, 18 November 2021
• 09:00 - 17:00 No lectures on Thu 18 Nov 2021
• Friday, 19 November 2021
• 14:00 - 16:30 Symbolic dynamics for nonuniformly hyperbolic systems  Convener: Yuri Lima (UFC, Brazil)
• Monday, 22 November 2021
• 09:00 - 17:00 No lectures on Mon 22 Nov 2021
• Tuesday, 23 November 2021
• 14:00 - 16:30 Symbolic dynamics for nonuniformly hyperbolic systems  Convener: Yuri Lima (UFC, Brazil)
• Wednesday, 24 November 2021
• 14:00 - 16:30 SRB Measures and Young Towers  Convener: José Ferreira Alves (University of Porto, Portugal)
• Thursday, 25 November 2021
• 14:00 - 16:30 Symbolic dynamics for nonuniformly hyperbolic systems  Convener: Yuri Lima (UFC, Brazil)
• Friday, 26 November 2021
• 14:00 - 16:30 SRB Measures and Young Towers  Convener: José Ferreira Alves (University of Porto, Portugal)
• Tuesday, 30 November 2021
• 14:00 - 16:30 Symbolic dynamics for nonuniformly hyperbolic systems  Convener: Yuri Lima (UFC, Brazil) Material:
• Wednesday, 1 December 2021
• 14:00 - 16:30 SRB Measures and Young Towers  Convener: José Ferreira Alves (University of Porto, Portugal)
• Friday, 3 December 2021
• 14:00 - 16:30 SRB Measures and Young Towers  Convener: José Ferreira Alves (University of Porto, Portugal) Material:
• Monday, 6 December 2021
• 14:00 - 16:30 Workshop
• 14:00 Young towers for geodesic flows on certain non-positively curved surfaces 1h15'
In this talk, based on joint work with Lima and Melbourne, we discuss how the ideas of Chernov for the construction of Young towers for piecewise smooth hyperbolic systems can be explored to yield similar structures for the geodesic flows on certain non-positively curved surfaces. In particular, by combining this fact with the results by Balint, Butterley, Melbourne, Terhesiu, Torok (among others), we derive several statistical consequences for this class of geodesic flows (including an effective rate of mixing and a central limit theorem).
 Speaker: Carlos Matheus (CNRS, France)
• 15:15 Sinai-Ruelle-Bowen measures for surface diffeomorphisms 1h15'
For a surface diffeomorphism f I will describe some natural geometric conditions that guarantee existence of a Young tower for f. The base of the tower is a rectangle'' which is maximal'' with respect to T-returns for some T and the inducing time of the tower is the T-return to its base. As a corollary one obtains existence of an SRB measure for f. This is a joint work with Vaughn Climenhaga and Stefano Luzzatto.
 Speaker: Yakov Pesin (Penn State, USA)
• Tuesday, 7 December 2021
• 14:00 - 16:30 Workshop
• 14:00 Dynamics of smooth surface diffeomorphisms: entropy continuity of exponents 1h15'
The top Lyapunov exponent and the entropy are well-known to depend discontinuously on the invariant measures. We show that the two defects in lower semicontinuity are linked for $C^\infty$ smooth surface diffeomorphisms. In particular, for a given sequence of ergodic measures, continuity of the entropy implies that of the exponent. The proof relies on Yomdin theory and a key reparametrization lemma for curves near homoclinic  tangencies. This has a number of consequences.

This is joint work with Sylvain CROVISIER and Omri SARIG.
 Speaker: Jerome Buzzi (Un Paris-Saclay, France)
• 15:15 Mixing rates for symplectic almost Anosov maps 1h15'
Establishing sharp bounds on the mixing rates of non-uniformly hyperbolic maps has been an active area of research for some time and Markov partitions in one way or another play an important role in this area. In this talk I will focus on a class of 2D, (non-uniformly) hyperbolic, symplectic (area preserving) maps that exhibit intermittent behaviour near a fixed point. The systems under considerations are examples of what are called "almost Anosov" maps, but in contrast to examples considered previously, in our examples, the derivative of the maps at the neutral fixed point is not the identity matrix and the stable and unstable distributions are tangent. This is the generic behaviour near a neutral fixed point for a 2D symplectic map and such a behaviour complicates considerably the geometrical analysis involved in estimating the mixing rate. I will briefly explain the class of examples and describe the ingredients that lead to proving sharp mixing rates for such systems.
(Joint work with Carlangelo Liverani)
 Speaker: Peyman Eslami (Un. Rome, Italy)
• Wednesday, 8 December 2021
• 14:00 - 16:30 Workshop
• 14:00 Markov partition for hyperbolic systems with singularities 1h15'
For the hyperbolic systems with singularities, Markov partitions are rather delicate to construct because of the fragmentation of the phase space by singularities. In this talk, we investigate a broad class of uniformly hyperbolic systems with singularities, which includes the Sinai dispersing billiards and their small perturbations due to external forces and nonelastic reflections with kicks and slips. Under the standard hypotheses (H1)-(H4), we establish countable Markov partitions by constructing perfect'' hyperbolic product sets with exponential tail. Such structure immediately implies the exponential decay of correlations, the central limit theorem and other advanced stochastic properties. We also obtain Markov partitions for certain nonuniformly hyperbolic systems, which includes the semi-dispersing billiards and the Bunimovich billiards. This is a joint work with Fang Wang and Hong-Kun Zhang.

 Speaker: Jianyu Chen (Soochow University, China)
• 15:15 Dynamics of smooth surface diffeomorphisms: strong positive recurrence 1h15'
The goal is to introduce and discuss the strong positive recurrence property for diffeomorphisms of a manifold.
I will show that every C-infinity surface diffeomorphism with positive topological entropy satisfies this property. The proof uses results presented by Jérôme Buzzi in his talk on the continuity of the entropy and exponents. Some consequences of this property will be discussed by O. Sarig later.
 Speaker: Sylvain Crovisier (Un. Paris_Saclay, France)
• Thursday, 9 December 2021
• 14:00 - 16:30 Workshop
• 14:00 Dynamics of smooth surface diffeomorphisms: Spectral gap and stochastic properties 1h15'
This talk is a continuation of the lectures by  Buzzi and Crovisier on  surface diffeomorphisms which are differentiable infinitely many times.

(1) Jerome proved that such diffeos exhibit "entropy continuity of Lyapunov exponents"

(2) Sylvain showed that entropy continuity of Lyapunov exponents implies a property we call "strong positive recurrence"

(3) I will show that "strong positive recurrence" implies exponential decay of correlations and CLT for mixing  measures of maximal entropy.

The method is to build a countable Markov partition whose transition matrix acts with spectral gap on  a nice Banach space. (Joint work with Buzzi & Crovisier)
 Speaker: Omri Sarig (Weizman, Israel)
• 15:15 Inducing schemes and their role in thermodynamics 1h15'
In this talk I will focus on systems which display certain hyperbolic features.
Inspired by the work of Lai-Sang Young '98 on constructing SRB measures,
Pesin, Senti, and Zhang '16 introduced maps with inducing schemes of hyperbolic type and effected thermodynamical formalism for such maps, i.e., constructed equilibrium measures for a broad class of potentials.
Particular examples of maps admitting inducing schemes of hyperbolic type include: the H\'enon map at first bifurcation,  the Katok map, and a slow-down of a classical Smale-Williams solenoid. The aim of this talk is to present a comprehensive view on the subject and to describe some new results regarding ergodic properties of such systems, such as decay of correlations, the Central Limit Theorem, the Bernoulli property, etc. This is a joint work with Farruh Shahidi.
 Speaker: Agnieszka Zelerowicz (U. of Maryland, USA)
• Friday, 10 December 2021
• 14:00 - 16:30 Workshop
• 14:00 On the thermodynamical formalism for expanding measures 1h15'  Speaker: Vilton Pinheiro (UFBA, Brazil) Material:
• 15:15 Symbolic dynamics for orbits with 0 Lyapunov exponents 1h15'
We introduce a class of orbits which do not require hyperbolicity. That is, sensitivity to initial conditions may be strictly sub-exponential. We discuss symbolic dynamics of such orbits (namely countable Markov Partitions with a finite-to-one almost everywhere coding, and with summable variations for the C^1 distance of stable and unstable manifolds of chains). We discuss a class of examples where every invariant measure can be coded this way, including the Riemannian volume Vol, while Vol-a.e point has a 0 Lyapunov exponent.
 Speaker: Snir Ben Ovadia (Weizman, Israel)