**RESEARCH IN RESIDENCE **

**
**#### Random ergodic optimization

#### Random ergodic optimization

*Optimisation ergodique aléatoire*

#### 30 June – 11 July, 2025

Ergodic optimization is a general framework in dynamical system where an observable is understood through its minimum over the set of invariant probability measures. A measure that attained that minimum is called minimizing. The general question is to understand the set of minimizing measures and more generally the set of minimizing orbits. One possible approach is to see minimizing measures as ground states, that is as limits of Gibbs measures at zero temperature. Gibbs measures are eigensolutions of the Ruelle operator or the transfer operator. We intend to study such questions in the framework of skew products of dynamical systems where the base dynamics is random (ergodic and invertible) and the fiber dynamics is a composition of maps indexed by a random environment. The problem of optimizing an observable in a random environment is new and has not yet been studied.

#### Participants

Eduardo Garibaldi (University of Campinas)

Philippe Thieullen (Université de Bordeaux)

Matias Zimmermann (University of Campinas)

*Ergodic optimization is a general framework in dynamical system where an observable is understood through its minimum over the set of invariant probability measures. A measure that attained that minimum is called minimizing. The general question is to understand the set of minimizing measures and more generally the set of minimizing orbits. One possible approach is to see minimizing measures as ground states, that is as limits of Gibbs measures at zero temperature. Gibbs measures are eigensolutions of the Ruelle operator or the transfer operator. We intend to study such questions in the framework of skew products of dynamical systems where the base dynamics is random (ergodic and invertible) and the fiber dynamics is a composition of maps indexed by a random environment. The problem of optimizing an observable in a random environment is new and has not yet been studied.*

#### SPONSOR