Nearly a hundred years ago Hadamar used infinite sequences of symbols to study the distribution of geodesics on certain surfaces, something which was also done in the 1930ies and 40ies by Hedlund and Morse. They started the study of sequence spaces with the shift operator as dynamical systems; they called this new field symbolic dynamics. Also in the 1940ies Shannon used sequence spaces to describe information channels. Since then, symbolic dynamics has been used in ergodic theory, topological dynamics, information theory, and in hyperbolic and complex dynamics. Of course, using infinite stings of symbols is much older … just think of the (decimal or binary) expansion of real numbers, the strings of “holes” on your cd’s and dvd’s, or the string of letters making up this text.
In this course we will study the basic properties of symbolic dynamics, and also focus on applications in other fields, such as number theory, and information theory.
The goal of this course is that students get familiar with studying and presenting (both by writing a small paper/lecture notes, and orally) a for them – new part of mathematics, which usually can only be known by studying research papers or the research monograph of Lind and Marcus.
An introductory lecture will be given by Kraaikamp, all other lectures by the participants. To do this, each participant will choose a subject from the list of subjects presented at the beginning of the course, and then – helped by Kraaikamp – will prepare lecture notes for a two times 45 minutes class on the chosen subject.
Literature and Study Materials
Apart from research papers, this course will be based on the book by Lind and Marcus: Lind, Douglas; Marcus, Brian: An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 1995. xvi+495 pp. ISBN: 0-521-55124-2; 0-521-55900-6
Additional study material
Kitchens, Bruce P.: Symbolic dynamics. One-sided, two-sided and countable state Markov shifts. Universitext. Springer-Verlag, Berlin, 1998. x+252 pp. ISBN: 3-540-62738-3. Boyle, Mike Algebraic aspects of symbolic dynamics. Topics in symbolic dynamics and applications (Temuco, 1997), 57—88, London Math. Soc. Lecture Note Ser., 279, Cambridge Univ. Press, Cambridge, 2000. Hao, Bai-Lin; Zheng, Wei-Mou: Applied symbolic dynamics and chaos. Directions in Chaos, 7. World Scientific Publishing Co., Inc., River Edge, NJ, 1998. xvi+443 pp. ISBN: 981-02-3512-7. An important, but not-so-easy book is that by Schmidt: Schmidt, Klaus: Dynamical systems of algebraic origin. Progress in Mathematics, 128. Birkhäuser Verlag, Basel, 1995. xviii+310 pp. ISBN: 3-7643-5174-8.
Assessment
In order to pass the exam for this course, each participant is asked to give at least one presentation, and to write one (or more) `paper’ (in LaTeX) on the material presented. This paper serves as an hand-out to the other participants. At the end of the course these hand-outs will form lecture notes on symbolic dynamics and related subjects. A typical lecture will consist of (part of) the book by Lind and Marcus, or a paper from the literature.
Remarks
This course is offered in Leiden, spring semester 2010. This course is offered every other year, so it will NOT be offered in 2010/11.