MATS530 MA1: Dimension Theory of Smooth Dynamical Systems (JSS25), 2 ECTS [home page]
Department of Mathematics and Statistics
You cannot register for the course because the course has expired.
The registration deadline for this course passed 30.4.15 at 23:59.
General information
Home page:  http://www.jyu.fi/summerschool 

Begins  ends:  17.8.15  21.8.15 
Registration period:  1.2.15 at 0:00
 30.4.15 at 23:59
The registration may be cancelled before 25.6.2015 at 00:00. 
Instructor(s):  Antti Käenmäki 
Credits:  2 ECTS cr. 
Languages:  language(s) of instruction: English; completion language(s): English 
Registered:  37 
Organisations:  Department of Mathematics and Statistics (MATHS), Faculty of Mathematics and Science (SCI), Jyväskylä Summer School (JSS), Mathematics (MAT) 
Current events: 

Contents:  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 lowdimensional uniformly hyperbolic systems. The basic concepts of multifractal analysis will be explained. Thereafter nonuniformly 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, MargulisRuelle formula, Pesin formula and LedrappierYoung formula. Finally we will give the ideas to show the exactdimensionality of hyperbolic measures. During the course we will indicate unsolved problems and areas of current research. 
Prerequisites:  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 higherdimensional real analysis will be useful. 
Modes of study:  Obligatory attendance at lectures, and completing the exercises. 
Completion mode:  Pass/fail 
Course workload:  10 h lectures (2 x 45 min/day, and consultation/day) 
[Limit information on study groups]
Lecture [group details and registration]
Location  Week  Day  Date  At  Supervisor  Further information  URI  Tapahtuman tiedot  

1  MaD 202  34  Mo  17.8.2015  10:0012:00  Käenmäki  Tapahtuman tiedot  
2  MaD 202  34  Tu  18.8.2015  10:0012:00  Käenmäki  Tapahtuman tiedot  
3  MaD 202  34  Wed  19.8.2015  10:0012:00  Käenmäki  Tapahtuman tiedot  
4  MaD 202  34  Th  20.8.2015  10:0012:00  Käenmäki  Tapahtuman tiedot  
5  MaD 202  34  Fr  21.8.2015  10:0012:00  Käenmäki  Tapahtuman tiedot 