MA1: Dimension Theory of Smooth Dynamical Systems
calendar_month 29 Iul 2015, 00:00
Dimension theory is a very useful tool to analyze the complicated structure of invariant sets or measures under a smooth chaotic system. While in general the analysis of invariant sets is not yet satisfactory the theory of invariant (hyperbolic) measures is relatively far developed. This makes it possible to study a large class of dynamical systems. The course will start by introducing basics of ergodic theory, existence of invariant measures (Birkhoff's Ergodic Theorem, ergodic decomposition and Oseledec' ergodic theorem). Following an introduction to fractal dimensions this will be applied to low-dimensional uniformly hyperbolic systems. The basic concepts of multifractal analysis will be explained. Thereafter non-uniformly hyperbolic systems in arbitrary dimension will be introduced. The fundamental concepts (Pesin theory) will be studied. These include the theory of Lyapunov exponents, local entropy, stable foliations, Margulis-Ruelle formula, Pesin formula and Ledrappier-Young formula. Finally we will give the ideas to show the exact-dimensionality of hyperbolic measures. During the course we will indicate unsolved problems and areas of current research.

Course leader
Lecturer: Prof. Jrg Schmeling (University of Lund, Mathematical Center, LTH, Sweden)Coordinator: Dr. Antti Kenmki (University of Jyvskyl, Finland)

Target group
The course is aimed at graduate students, but strong advanced undergraduate students with the appropriate background might find it suitable. Basic knowledge on measure/integration theory and higher-dimensional real analysis will be useful. The Summer School annually offers courses for advanced master's students, graduate students, and post-docs in the various fields of science and information technology.

Course aim
The most important aims of the Summer School are to develop post-graduates scientific readiness and to offer students the possibility to study in a modern, scientific environment and to create connections to the international science community.

Credits info
2 ECTS Passing: Obligatory attendance at lectures, and completing the exercises.Grading: Pass/fail

Fee info
EUR 0: Participating the Summer School is free of charge, but student have to cover the costs of own travel, accommodation and meals at Jyvskyl.

The 25th Jyvskyl Summer School is not able to grant any Summer School students financial support. In order to ensure your participation, we recommend that you take steps to secure your own funding, for example, by turning first to your home institution

University of Jyv%C3%A4skyl%C3%A4
Address: Faculty of Mathematics and Science and Faculty of Information Technology, Jyv%C3%A4skyl%C3%A4 Summer School, Faculty of Mathematics and Science P.O.Box 35 (YK312), FIN-40014 University of Jyv%C3%A4skyl%C3%A4, Finland
Postal code: FIN-40014
City: Jyv%C3%A4skyl%C3%A4
Country: Finland
Phone: +358505818351