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.
Registration:
Please register here to join the zoom meeting and to watch the lecture videos.
Lecturers:
Stefano Galatolo (University of Pisa, Pisa, Italy)
Isaia Nisoli (UFRJ, Rio de Janeiro, Brazil / Hokkaido University)
Yuzuru Sato (Hokkaido University)
Date:
1-5 August, 2022
Place:
3-204, Department of Mathematics, Hokkaido Unviersity
Program: [Lecture videos will be available on 5 August, 2022]
Monday, 1 August 17:00-20:00 (JST) / 10:00-13:00 (CET)
17:00-17:20 Introduction: "Nonlinear stochastic phenomena in random dynamical systems." by Yuzuru Sato [video]
17:20-18:30 Lecture 1: "Chaos, the statistical properties of dynamics, invariant measures, ergodic theorems." by Stefano Galatolo [video]
18:40-20:00 Lecture 2 "Introduction to Julia. Numerical experiments on the statistical properties of dynamics." by Isaia Nisoli Jupyter Notebook: Lecture 1 [video]
20:00-21:00 Open discussions at spatial chat discussion room by YS, SG, IN
Tuesday, 2 August 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" by SG [video]
18:40-20:00 Lecture 4: "Approximating the transfer operator with a computer: the Ulam Method, interval arithmetics" by IN Jupyter Notebook: Lecture 2 [video]
20:00-21:00 Open discussions at spatial chat discussion room by YS, SG, IN
Wednesday, 3 August 17:00-20:00 (JST) / 10:00-13:00 (CET)
17:00-18:30 Lecture 5: " Quantitative stability results, rigorously validating the numerical approximation" by SG [video]
18:40-20:00 Lecture 6: "Rigorous computations, computing the invariant measure and Lyapunov exponent with a controlled error" by IN Jupyter Notebook: Lecture 3 [video]
20:00-21:00 Open discussions at spatial chat discussion room by YS, SG, IN
Thursday, 4 August 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". by SG [video]
18:40-20:00 Lecture 8: "The approximation of the stationary measure and Lyapunov exponents with a controlled error in the random case". by IN Jupyter Notebook: Lecture 4 [video]
20:00-21:00 Open discussions at spatial chat discussion room by YS, SG, IN
Download the notebook, complete the julia program, and run it to compute Lyapunov exponents. Send the results to ysato_at_math.sci.hokudai.ac.jp until Friday. If you have questions, please join us for discussion at spatial chat at 5pm (JST) on Friday, 5 August.
Friday, 5 August 17:00-20:00 (JST) / 10:00-13:00 (CET)
17:00-20:00 Open discussions at spatial chat discussion room by YS, SG, IN
Lecture notes and other resources:
S. Galatolo "Statistical properties of dynamics. Introduction to the functional analytic approach"
Related papers:
S. Galatolo, M. Monge, I. Nisoli Existence of Noise Induced Order, a Computer Aided Proof Nonlinearity, 33(9):4237--4276, (2020).
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
S. Galatolo, I. Nisoli, An elementary approach to rigorous approximation of invariant measures. SIAM J. Appl. Dyn. Syst. 13 (2014), no. 2, 958--985.
S. Galatolo, , I. Nisoli, 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.
Organizer:
Yuzuru Sato (Hokkaido University) Email: ysato_at_math.sci.hokudai.ac.jp
Isaia Nisoli (UFRJ, Brazil / Hokkaido University) Email: isaia.nisoli_at_es.hokudai.ac.jp
Y.S is supported by JSPS Grant-in-Aid for Scientific Research (B) No. 17H02861, 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.