An introduction to fluctuations of observables in ergodic theory

Jean-Rene Chazottes (Ecole Polytechnique, Paris)

The main topics of these lecture notes will be:

- Birkhoff ergodic theorem and why it doesn't make sense to speak of speed of convergence in this theorem; nevertheless there is the shrinking theorem by Ivanov;

- mixing and decay of correlations, central limit theorem (also almost-sure invariance principle, almost-sure central limit theorem);

- large deviations;

- and concentration inequalities.

Usually, observations of a dynamical system are made by taking the Birkhoff average of a function on phase space, but there are many other observables which are not Birkhoff averages (we shall see examples). This is precisely the scope of concentration inequalities to handle quite general observables, as well as to get non-asymptotic results.

In these lectures, a guiding example will be uniformly expanding maps of the interval, but without a Markov partition. This will show the power of the transfer operator method in getting limit theorems.

An Introduction to stochastic stability for discrete dynamical systems

Vitor Araujo (UFRJ, Rio de Janeiro)

This will be an introduction to the idea of random perturbations, stationary measures and the concept of stochastic stability, with a (sketch of the) proof of stochastic stability for uniformly expanding maps.

I plan to start with an explanation of what a stationary probability is and then the idea of stochastic stability, exemplifying with uniformly expanding maps and different perturbations schemes. Then I will introduce entropy, pressure and equilibrium states, and show that under some conditions (satisfied for example by 2x (mod1)) some equilibrium states vary continuously for small random perturbations. As an application we derive the stochastic stability of uniformly expanding maps and (perhaps) also of a class of non-uniformly expanding maps.

This is the basic outline. It will be mostly the "entropy, pressure and equilibrium states" part of an ergodic theory course plus some interaction with random perturbations, markov chains and stationary densities for uniformly expanding systems.

An introduction to entropy

Alexander Bufetov (Rice University, USA)

This mini-course will be devoted to ergodic-theoretic entropy.

Originating in the work of Clausius and Boltzmann in statistical

mechanics, entropy was introduced to coding theory by Shannon and to ergodic theory by Kolmogorov. We shall start with a brief exposition of Shannon entropy, proceed to Kolmogorov-Sinai entropy, and finish with a survey of B.M. Gurevich's work on the entropy of symbolic flows.  The course does not require any prior knowledge of ergodic theory.

Programme:

1. The Shannon Entropy. The Kraft-Macmillan Inequality. The Huffman Encoding.

2. The Kolmogorov-Sinai Entropy. Examples. The Kolmogorov-Sinai Theorem.

3. The Shannon-Macmillan-Breiman Theorem. The Abramov Formula. Entropy of

Symbolic Flows.

An introduction to random perturbation in continuous time

Greg Pavliotis, Imperial College London

This course will be a (very) brief introduction to stochastic differential equations (SDEs) and to some of the qualitative properties of their solutions; in particular, the long time/weak noise asymptotic behaviour of solutions to SDEs will be investigated. We will start by presenting some basic properties of SDEs (definition of Brownian motion, definition of an SDE, basic existence and uniqueness theorem). We will then discuss briefly about the ergodic properties SDEs. Then, we will study in some detail SDEs with periodic coefficients (i.e., SDEs on the unit torus) and we will show how the presence of noise leads to the uniqueness of the invariant measure for the SDE. Finally, we will study the long time, weak noise asymptotics for one dimensional Hamiltonian systems with a periodic potential, perturbed by dissipation and noise. We will show, in particular, the "universality" of the long time/weak noise asymptotics for this system.