Minicourses

Ergodic optimization, ergodic dominance, and thermodynamic formalism

by Oliver Jenkinson

I will start with the History, examples, connections with thermodynamic formalism. After I will talk about the "coboundary trick", generic properties of maximizing measures, and the dual problem of finding a continuous function maximised by a given measure. At the end I will give more examples, leading to connections with stochastic dominance.

Ergodic Transport

by Jairo Mengue and Rafael Rigão Souza

We will present some recent results in which ideas from Classical Transport Theory are applied to Ergodic Theory. The dynamical Kantorovich duality will be one of the main topics we will describe. Some of these results are natural generalizations of the ones usually considered in Classical Ergodic Optimization.

Multiplicative ergodic theorems and applications

by Anthony Quas

In 1965, Oseledets proved the landmark Multiplicative Ergodic Theorem, describing the behaviour of the composition of a stationary sequence of linear maps of a finite-dimensional vector space. This has immediate applications to differentiable dynamical systems, and leads directly to the study of non-uniform hyperbolicity. The Multiplicative Ergodic Theorem has a wide range of applications in other areas also. The theorem has been extended in many directions, to operators on Banach spaces and abstractly to non-expanding maps of geometric spaces. In this course, we will see the connection with the Kingman sub-additive ergodic theorem, give outlines of a proof of the MET and describe the Banach space extensions. We will also describe recent work extending the Oseledets multiplicative ergodic theorem to the semi-invertible context (where the operators are non-invertible, but the underlying base dynamics is invertible). This extension has been undertaken with a view to applying the results in collaboration with atmospheric scientists, in a way which I will talk about.

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