# Online lectures series: 'Introduction to computational ergodic theory'.

# Online lectures *“Introduction to computational ergodic theory”*

## 1-5 August, 2022

The Hokkaido Summer Institute organizes a zoom online lecture series on computational ergodic theory on 1-5 August, 2022. The goal of the lectures is to introduce a software and the related theory providing tools for computer aided proofs and rigorous estimates on the statistical properties of dynamics. The lectures includes basics on ergodic theory, tutorials on the language Julia, with an introduction to interval arithmetic libraries and computer assisted proofs in dynamical systems and ergodic theory problems and the package RigorousInvariantMeasure.jl, now part of the JuliaDynamics organization https://github.com/JuliaDynamics/RigorousInvariantMeasures.jl

Registration will be required to attend the zoom online lectures at https://sites.google.com/view/hsi-comp-ergo-theo-2022/ The lectures will be recorded and will be published after the school.

### “Introduction to computational ergodic theory”

Ergodic theory is a mathematical field that connects dynamical system theory and probability theory, and underlies statistical mechanics and nonlinear physics. However, mathematically, it is difficult to analyze even basic problems such as the uniqueness and existence of invariant densities. One of the purposes of this lectures is to learn how a functional analytic approach can be used for these kind of problems in dynamical systems, giving an efficient method to get information on the invariant measures and on the statistical behavior of the system. The second purpose is to introduce the recently developed validated numerics for computer-assisted proofs in dynamical systems and ergodic theory and to understand the results of proof of existence of noise-induced phenomena in nonlinear physics.

Date: 1-5 August, 2022

Place: 4-501, Department of Mathematics, Hokkaido Unviersity

Lecturers: Stefano Galatolo (University of Pisa, Pisa, Italy) and Isaia Nisoli (UFRJ, Rio de Janeiro, Brazil / Hokkaido University)

Program:

Monday, 1 August 17:00-20:00 (JST) / 10:00-13:00 (CET)

Lecture 1: “Chaos, the statistical properties of dynamics, invariant measures, ergodic theorems.”

Lecture 2 “Introduction to Julia. Numerical experiments on the statistical properties of dynamics.”

Tuesday, 2 August 17:00-20:00 (JST) / 10:00-13:00 (CET)

Lecture 3: “The transfer operator approach, statistical properties of expanding maps and other examples”

Lecture 4: “Approximating the transfer operator with a computer: the Ulam Method, interval arithmetics”

Wednesday, 3 August 17:00-20:00 (JST) / 10:00-13:00 (CET)

Lecture 5: " Quantitative stability results, rigorously validating the numerical approximation”

Lecture 6: “Rigorous computations, computing the invariant measure and Lyapunov exponent with a controlled error”

Thursday, 4 August 17:00-20:00 (JST) / 10:00-13:00 (CET)

Lecture 7: “Random dynamical systems, the transfer operator approach and the quantitative statistical stability”.

Lecture 8: “The approximation of the stationary measure and Lyapunov exponents with a controlled error in the random case”.

Friday, 5 August 17:00-20:00 (JST) / 10:00-13:00 (CET)

Open discussions at spatial chat.