Objectifs :
The aim of this minicourse is to introduce the ergodic optimization and its application in Dynamical systems, Spectral Theory, Thermodynamical Formalism and Multifractal Analysis. Moreover, there will be provided insights about current research directions in the ergodic optimization.

Programme :
1- Fundamentals
- Introduction to dynamical systems
- Introduction to ergodic theory
- Introduction to thermodynamic formalism
- Introduction to the ergodic optimization of Birkhoff averages
2- Lyapunov exponent of matrix cocycles
- Definition of subadditive and almost additive potentials
- Definition of Linear cocycles of matrix cocycles and examples of it
- Existence of Lyapunov exponents: Furstenberg and Kesten Theorem, The Kingman's subadditive theorem, the Kifer's example, ...
- Generic matrix cocycles and its properties
- Introduction to the ergodic optimization of Lyapunov exponents
3- Subadditive thermodynamic formalism
- Varitional principle for subadditive potentials
- Definition of equilibrium and Gibbs measures for subadditive potentials
- Uniqueness of equilibrium measures
4- Some notes on advanced topics
- Zero temprature limits for subadditive potential
- Restricted variatinal principle for generic matrix cocycles

Pré-requis :
- Working knowledge of measure theory and functional analysis.
- Probability I,II

Compétences acquises à l'issue de la formation :
A student
- gets a good knowledge of dynamical systems
- learns the main differences appearing between the ergodic optimization of Birkhoff averages and the ergodic optimization of Lyapunov exponents by using some basic examples
- understands the existence of a cone filed implies matrix cocycles are almost additive.
- knows about some open problems