19216801

SoSe 15: Ergodentheorie und Transferoperatoren

Peter Koltai

Hinweise für Studierende

Prüfungsmodus: Klausur

Zusätzl. Angaben / Voraussetzungen

Zielgruppe:
Masterstudenten, oder Fortgeschrittene Bachelorstudenten

Voraussetzungen:
Maßtheorie und grundlegende Kenntnisse aus der Funktionalanalysis. Von Vorteil, aber nicht notwendig: grundlegende Kenntnisse aus der Theorie dynamischer Systeme (z.B. die Vorlesung Differentialgleichungen I) Schließen

Kommentar

Description:
Ergodic theory is concerned with the behavior of dynamic systems when these are running for a long time. Vaguely speaking, the long-term statistical behavior of an ergodic dynamical system is not going to depend on its initial condition. This course discusses the mathematical characterization of this property. A central role is going to be played by the so-called transfer operator, which describes the action of the dynamics on a distribution of states. We are also going to higlight its importance in applications, when it comes to the numerical approximation of quantities of interest. If time permits, we will introduce entropy, as a notion of complicatedness (or "non-predictability") of a dynamical system.

Planned content

Basic Constructions:
measure preserving transformations, ergodicity and mixing, the Poincaré Recurrence Theorem, stochastic processes as deterministic dynamical systems, ergodicity and mixing, Markov chains, skew products
Ergodic Theorems:
Perron-Frobenius theory, von Neumann's Mean Ergodic Theorem, Birkhoff's Pointwise Ergodic Theorem, Kingman's Subadditive Ergodic Theorem (measure disintegration, ergodic decomposition)
Spectral Theory:
the Koopman Operator and the spectral approach to ergodic theory
Transfer Operators:
the Perron-Frobenius Operator, quasi-compactness, absolutely continuous invariant measures, Ulam's method
Entropy: (content covered here depending on time left) information content, entropy of a partition, metric entropy, topological entropy.
Applications:
Oceanography, atmospheric and molecular dynamics, internet search.

Zur Vorlesungshomepage Schließen

Literaturhinweise

Literatur:
Primär:

Omri Sarig; Lecture Notes on Ergodic Theory. Online Erreichbarkeit: www.wisdom.weizmann.ac.il/~sarigo/506/ErgodicNotes.pdf

Begleitend:

Peter Walters; An Introduction to Ergodic Theory. Springer, 1982
Andrzej Lasota and Michael C. Mackey; Chaos, Fractals, and Noise. Springer, 1994
Michael Brin and Garrett Stuck; Introduction to Dynamical Systems. Cambridge University Press, 2003
Abraham Boyarsky, Pawel Góra; Laws of Chaos. Springer Science+Business Media New York, 1997

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