Simons Semester in Banach Center: "Dynamical Systems"

Jörg Schmeling: Dimensional aspects in smooth dynamical systems

Jörg Schmeling (University of Lund)

Dimensional aspects in smooth dynamical systems

8 x 45min


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.


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.