In this course, we firstly study the fundamentals of topological dynamical systems on Cantor sets. As the Cantor space is homeomorphic with any compact space arisen from countable (infinite) product of a finite alphabet, there are rich sources of combinatorics supporting various properties of the dynamical systems on them that we are intending to introduce and study some of them. Besides, some algebraic tools associated to the dynamics on the Cantor sets as well as applications of minimal Cantor systems in the theory of growth of groups will be presented.

Syllabus:

1. Topological dynamical systems on Cantor Spaces, Subshifts
2. Return words, Rauzy Graphs, Primitive Substitutions and S-adic subshifts, Recognizability
3. Some Motivations for Studying Minimal Dynamics (interval exchange transformations)
4. Examples (Substitutions, Toeplitz Shifts)
5. Partitions in Towers
6. Ordered Groups and Direct limits associated to a sequence of K-R Partitions
7. Dimension Group and Measures
8. Ordered Cohomology Group of Cantor Minimal systems
9. Full Groups and Topological Full Groups of Cantor Minimal Systems
10. Simplicity of the derived subgroup of the topological full group
11. Algebraic structure of the full group : amenability and LEF property.

Prerequisite:

   An introductory course in topological dynamics and ergodic theory.


References:

1. F. Durand, D. Perrin, Dimension Groups and Dynamical Systems, preprint.
2. E.G. Effros, Dimensions and C*-algebras, AMS, 1981.
3. P. Kurka, Topological and Symbolic Dynamics, Soc. Math France, 2003.
4. D. Lind and B. Marcus, An introduction to symbolic dynamics and coding
5. K. Juschenko, A companion to the mini-course on full topological groups of Cantor minimal. pdf file
6. M. Queffelec, Substitution Dynamical Systems - Spectral Analysis, 2010.
7. Y. De Cornulier, Groupes pleins-topologiques (d’aprs Matui, Juschenko, Monod,..). Astérisque No. 361 (2014), Exp. Bourbaki No. 1064, pp. 183-223.

Grading Scheme:

   Exercises: 50%
   Final Projects: 50%