Hokkaido University Summer Institute: Introduction to dynamical systems

2017-7-31 13:30 - 2017-8-3 16:15
4-501, Faculty of Science, Hokkaido University

Part I:31 July, 1 August, 13:00-14:30, 14:45-16:15, Takao Namiki

The main topics of the lecture are the theory and application on dynamical systems and ergodic theory, especially on symbolic dynamics.

  • Basic notions on dynamical systems

  • Application: time-series analysis on brain wave data

  • Entropy, topological pressure and variational principle

  • Application: fractal geometry and cookie-cutter map

Part II: 2 August, 3 August, 13:00-14:30, 14:45-16:15, Stefano Galatolo

The satistical properties of a dynamical system can be well understood by the study of the associated transfer operator (considered on a suitable function space). In the minicourse following questions will be addressed:

  • existence of a regular invariant measure;

  • \item Lasota Yorke inequalities and spectral gap;

  • \item decay of correlations and some limit theorem;

  • \item stability under perturbations of the system.

The point of view taken is to present the general construction and ideas needed to obtain these results in the simplest way, avoiding technicalities. Plan of the lectures:

  1. Generalities on dynamical systems and ergodic theory. The transfer

  2. Regularizing action of the transfer operator on suitable function
    spaces. Lasota Yorke inequalities.

  3. Spectral gap and consequences. Limit theorems.

  4. Quantitative stability results, Linear response.