Thermodynamic formalism provides a mathematical framework for studying qualitative and quantitative aspects of dynamical systems. Founders of this theory include Sinai, Bowen, and Ruelle, but principle ideas have their origin in the development by Gibbs to describe interacting particle physical systems in the realm of statistical mechanics. Key concepts include Bernoulli and Gibbs distributions, equilibrium states, variational and, in particular, maximum entropy principles. Its contemporary mathematical aspects are intimately related with ergodic theory and have applications in fractal geometry, the description of large deviations, multifractal analysis, analysis of axiom A-systems, rigidity problems, ergodic optimization and control problems, among many others. There are also relations to geometry and group theory.
Classical approaches describe asymptotic behavior of Birkhoff (arithmetic) averages of Hölder continuous potentials and their fluctuations by means of pressure functionals, variational principles for ergodic measures and the analysis of maximizing (equilibrium) measures. This theory has experienced recently a series of extensions as so far it was not able to cover more general types of dynamical systems that are not uniformly hyperbolic or partially hyperbolic, randomized, or have a higher-dimensional phase space and hence require to deal with nonconformal systems.
Those extensions gave rise to what is nowadays called non-additive thermodynamical formalism. The origin of this notion is due to investigation of the growth of the norm (or a singular value function) of a matrix product. This line of investigations naturally occurs in the study of matrix cocycles. But it has also immediate impacts in the analysis of Lyapunov exponents and hence in the description of the lack of hyperbolicity, for example.
This area is developing very fast in the last time and in multiple directions. One common feature is that approaches in different areas have comparable flavors and common underlying ideas. In this meeting we want to gather specialists working from different perspectives. We hope that this meeting can provide a platform to create synergies and to give rise to many fruitful scientific collaborations.
Anke D. Pohl
Jairo Bochi (*)
Jérôme Buzzi (*)
Noé Cuneo (*)
Polina Vytnova (*)
(* to be confirmed)