Mathematics 388: An introduction to dynamical systems and ergodic theory
(Spring 2005)
Description
Dynamical systems is the study of the long-term behavior of any system,
mathematical or physical, that evolves over time. Ergodic theory is the
statistical study of a group of motions of a space with a measurable
structure on it. Both subjects evolved from efforts to understand
classical physical systems, and there is a lot of interface between the
two with much of ergodic theory applied to smooth dynamical systems.
In this course we will give a broad overview of the central topics in
dynamical systems and ergodic theory. The course is intended for
nonexperts who wish to learn the basic ideas and techniques in the
subject. Examples and applications will be highlighted throughout the
course. Rigorous coverage of the topics will allow those who wish to
pursue topics in more depth to proceed further right away, and
applications discussed will correspond to the interests of the audience.
The topics covered will include topological dynamics, topological
entropy, expansiveness, definitions of chaos, symbolic dynamics,
ergodicity and mixing properties, hyperbolic dynamics, Markov partitions,
and attractors with examples and (brief) applications in the areas of data
storage, search engines, cellular automata and number theory, complex
dynamics,
geodesic flows, and other physical applications depending on time and
interest.
Instructor
Hawkins
Schedule
MWF 11:55-12:45, Physics 120
Text
The main text will be Introduction to Dynamical Systems, by M. Brin
and G. Stuck. It is also a good reference book.
Prerequisites
The course will be largely self-contained assuming a
reasonable background in undergraduate analysis and basic topological
concepts. There will be a fast review of the measure theory and topology as
needed.
Reference(s)
-
Introduction to Dynamical Systems, by M. Brin and G. Stuck
-
An Introduction to Chaotic Dynamical Systems, by R. Devaney
-
Introduction to the Modern Theory of Dynamical Systems, A. Katok and B.
Hasselblatt
-
Ergodic Theory, K. Petersen
-
Mathematical Methods of Classical Mechanics (Graduate Texts in
Mathematics, No 60) , V.I. Arnold
Auditors/visitors are welcome. Check the course web page
http://www.math.duke.edu/~jhawkins/math388/math388.html for
upcoming topics, and feel free to come for periods of time to learn about
specific topics.
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