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MATS530 MA1: Dimension Theory of Smooth Dynamical Systems (JSS25), 2 ECTS [home page]

Department of Mathematics and Statistics

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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:
  • Lecturer: Prof. Jörg Schmeling (University of Lund, Mathematical Center, LTH, Sweden)
  • The course is a part of the 25th international Jyväskylä Summer School programme.

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.


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.

Modes of study:

Obligatory attendance at lectures, and completing the exercises.

Completion mode:


Course workload:

10 h lectures (2 x 45 min/day, and consultation/day)

[Limit information on study groups]

Lecture [group details and registration]

Lecture [group details and registration]; registered 37, maximum 200
reg.time: 1.2.2015 00:00 - 30.4.2015 23:59
 LocationWeekDayDateAtSupervisorFurther informationURITapahtuman tiedot
1MaD 20234Mo17.8.201510:00-12:00KäenmäkiTapahtuman tiedot
2MaD 20234Tu18.8.201510:00-12:00KäenmäkiTapahtuman tiedot
3MaD 20234Wed19.8.201510:00-12:00KäenmäkiTapahtuman tiedot
4MaD 20234Th20.8.201510:00-12:00KäenmäkiTapahtuman tiedot
5MaD 20234Fr21.8.201510:00-12:00KäenmäkiTapahtuman tiedot