Symbolic dynamics for nonuniformly hyperbolic systems 3 of 5
Channel: ICTP Mathematics
Date: 2021-11-23

Convener: Yuri Lima (UFC, Brazil)
Mini-Course
Markov Partitions and Young Towers in Dynamics | (smr 3642) 
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.