250095 VO Ergodic Theorie/Ergodentheorie II
Email
H. Bruin
for further information for this course.
Announcements
There will be no further lectures (i.e., with the
Shannon-Breiman-McMillan Theorem covered in early January
is the final topic of this course.
For the oral exam, please make an appointment by emailin due course.
The material is:
- Definition and properties of entropy (both topological and metric).
- Basic examples of computation of entropy (Bernoulli shifts, piecewise
linear interval maps.
- The variational principle.
- Basic notions and motivation of thermodynamic formalism.
- Applications of thermodynamic formalism to (fractal) dimension theory.
(including dimension formulas).
- The Shannon-Breiman-McMillan Theorem.
Schedule
| Day | Time | Room | | from | until |
|---|
| Wednesday | 14:00-15:45 | SR11 | Lecture | 03.3.2014 | 23.06.2014 |
| Wednesday | 14:00-14:45 | SR11 | Lecture | 04.3.2014 | 25.06.2014 |
Contents of the course
This course goes deeper into ergodic theory, with main content:
the theory of entropy and methods of thermodynamic formalism.
Topics to be discussed are likely to include
- Definition and properties of entropy (both topological and metric).
The variational principle.
- Entropy as invariant under isomorphism, and complete invariant under
isomorphism for Bernoulli shifts.
- The Shannon-Breiman-McMillan Theorem.
- Basic notions and motivation of thermodynamic formalism.
- Applications of thermodynamic formalism to (fractal) dimension theory.
Dimension formulas.
- Entropy and pressure for induced transformations (Abramov's formula).
The course will be given in English
References
-
Peter Walters, An Introduction to Ergodic Theory, Springer-Verlag 1975
ISBN 0-387-95152-0.
- Gerhard Keller, Equilibrium States in Ergodic Theory, Cambridge Univ. Press. 1998
- Rufus Bowen, Equilibrium states and ergodic Theory of Anosov Diffeomorphisms, Springer-Verlag, Lect. Notes in Mathematics, 470, 1975.
- Omri Sarig, Lecture Notes on Thermodynamic Formalism for Topological Markov Shifts. Penn State, 2009.
- Omri Sarig, Lecture Notes on Ergodic Theory
Penn State, Fall 2008,
in .pdf
and the later version:
in .pdf
Class Notes from earlier courses:
-
Class notes of Ergodic Theory I (SS2014) pdf,
updated November 4 2014.
This set of notes may still be updated, and corrected.
- Class notes of Thermodynamic Formalism I (WS2012) pdf.
Assessment
Will be based on an oral exam (in English by default, aber auf Deutsch ist auch möglich ).
The material is:
- Definition and properties of entropy (both topological and metric).
- Basic examples of computation of entropy (Bernoulli shifts, piecewise
linear interval maps.
- The variational principle.
- Basic notions and motivation of thermodynamic formalism.
- Applications of thermodynamic formalism to (fractal) dimension theory.
(including dimension formulas).
- The Shannon-Breiman-McMillan Theorem.
Updated January 2015