Zoom online lecture series, Hokkaido Summer Institute at Hokkaido University

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. 

Registration:

Please register here to watch the lecture videos.

Lecturers:

Stefano Galatolo (University of Pisa, Pisa, Italy)

Isaia Nisoli (UFRJ, Rio de Janeiro, Brazil)

Date:

6-10, September, 2021

Program:

Monday, 6 September 17:00-20:00 (JST) / 10:00-13:00 (CET)

17:00-18:30 Lecture 1: "Chaos, the statistical properties of dynamics, invariant measures, ergodic theorems."

18:40-20:00 Lecture 2 "Introduction to Julia. Numerical experiments on the statistical properties of dynamics."

20:00-21:00 Open discussions at spatial chat discussion room

Video: Introduction (Vimeo)

Video: Lecture 1 (Vimeo) Slides for Lecture 1 (Google Drive)

Video: Lecture 2 (Vimeo) Jupyter Notebook for Lecture 2 (Google Drive)

Tuesday, 7 September 17:00-20:00 (JST) / 10:00-13:00 (CET)

17:00-18:30 Lecture 3: "The transfer operator approach, statistical properties of expanding maps and other examples"

18:40-20:00 Lecture 4: "Approximating the transfer operator with a computer: the Ulam Method, interval arithmetics"

20:00-21:00 Open discussions at spatial chat discussion room

Video: Lecture 3 (Vimeo) Slides for Lecture 3 (Google Drive)

Video: Lecture 4 (Vimeo) Jupyter Notebook for Lecture 4 (Google Drive)

Wednesday, 8 September 17:00-20:00 (JST) / 10:00-13:00 (CET)

17:00-18:30 Lecture 5: " Quantitative stability results, rigorously validating the numerical approximation"

18:40-20:00 Lecture 6: "Rigorous computations, computing the invariant measure and Lyapunov exponent with a controlled error"

20:00-21:00 Open discussions at spatial chat discussion room

Video: Lecture 5 (Vimeo) Slides for Lecture 5 (Google Drive)

Video: Lecture 6 (Vimeo) Jupyter Notebook Lecture 6 (Google Drive)

Thursday, 9 September 17:00-20:00 (JST) / 10:00-13:00 (CET)

17:00-18:30 Lecture 7: "Random dynamical systems, the transfer operator approach and the quantitative statistical stability".

18:40-20:00 Lecture 8: "The approximation of the stationary measure and Lyapunov exponents with a controlled error in the random case".

20:00-21:00 Open discussions at spatial chat discussion room

Video: Lecture 7 (Vimeo) Slides for Lecture 7 (Google Drive) (A corrected version of the slides appearing in the video.)

Video: Lecture 8 (Vimeo) Jupyter Notebook for Lecture 8 (Google Drive)

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

18:00-20:00 Open discussions at spatial chat discussion room

Lecture notes and other resources:

S. Galatolo "Statistical properties of dynamics. Introduction to the functional analytic approach" (arXiv:1510.02615)

Related papers:

1. S Galatolo, M Monge, I Nisoli Existence of Noise Induced Order, a Computer Aided Proof Nonlinearity, 33(9):4237--4276, (2020).

2. S. Galatolo, I. Nisoli, B. Saussol. An elementary way to rigorously estimate convergence to equilibrium and escape rates. J. Comput. Dyn., 2015, 2 (1) : 51-64. doi: 10.3934/jcd.2015.2.51

3. Galatolo, Stefano; Nisoli, Isaia An elementary approach to rigorous approximation of invariant measures. SIAM J. Appl. Dyn. Syst. 13 (2014), no. 2, 958--985.

4. Galatolo, Stefano; Nisoli, Isaia Rigorous computation of invariant measures and fractal dimension for maps with contracting fibers: 2D Lorenz-like maps. Ergodic Theory Dynam. Systems 36 (2016), no. 6, 1865--1891.

5. Chihara, Takumi, Sato, Yuzuru, Nisoli, Isaia, and Galatolo, Stefano, "Existence of multiple noise-induced transitions in a Lasota-Mackey map," arxiv: 2102.11715, (submitted) (2021).

Organizer:

Yuzuru Sato (Hokkaido University) Email: ysato_at_math.sci.hokudai.ac.jp

Y.S is supported by JSPS Grant-in-Aid for Scientific Research (B) No. 17H02861, (C) No. 18K03441, and (B) No. 21H01002, London Mathematical Laboratory external fellowshp, Hokkaido Summer Institute Program, and Department of Mathematics / Research Institute for Electronic Sciences at Hokkaido University.

S.G. is partially supported by the research project PRIN 2017S35EHN_004 "Regular and stochastic behavior in dynamical systems" of the Italian Ministry of Education and Research.