Thematic Program on
 Dynamical Systems
  School of Mathematics, IPM,
 February - May, 2017
IPM
School of Mathematics
Geometry
 & Topology


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Mini Course
Speakers:   
 Dawoud Ahmadi Dastjerdi  (University of Guilan, Rasht, Iran)
Maryam Hosseini  (IPM, Tehran, Iran)
Title: An Introduction to Symbolic Dynamics

Slides: [1], [2]
Date &
Time:
 Tuesday, Jan. 31, 2017, 9:00--12:30
Wednesday, Feb. 1, 2017, 9:30--12:30 (Lecture Hall 1)
Thursday, Feb. 2, 2017, 9:30--12:30
Location: 
Lecture Hall 2,
IPM Niavaran Building,
Niavaran Square, Tehran
Poster
Description:
In this series of talks, subshifts in symbolic dynamics will be introduced. After giving the main definitions and some applications we continue in two basic areas of subshifts namely coded systems and minimals. These are the most studied in symbolic dynamics and show how multifaceted the routines could be.

An outline of subjects is as follows. Order may not necessarily be respected.

1. Basic definitions and applications of general subshifts.

2. Coded systems
(a) Subshift of finite type (SFT) and sofics,
i.  Adjacency matrix, Fischer cover and associated properties;
ii. Classification: equivalencies via dimension groups, Jordan forms, zeta function,  etc.
(b) Synchronized systems,
i.  Specified systems, almost sofic;
ii. Drive sets and depths.
(c) Beyond synchronized systems, mainly half synchronized systems.
(d) Generators and  relation with mixing properties when applicable.
(e) Examples, among others, $\beta$ and $S$-gap shifts.
3. Minimal systems,
(a) Some examples: Odometers, Toeplitz, Substitutions.
(b) Kakutani-Rokhlin towers, Bratteli diagram,
(c) Interaction between, shifts, topological dynamics and ergodic theory.

4. Dimension group, Automorphism group  (SFT's and Minimal's).

5. Some major problems in symbolic dynamics.

References: 
[1]   


[2]


[3]


[4]


[5]


[6]


[7]


[8]


[9]


[10]


[11]


[12]


[13]
F. Blanchard and G. Hansel, Systèmes codés, Theoretical Computer Science. 44 14-49, 1986.

M. Boyle, Topological Orbit Equivalence and Factor Maps in Symbolic Dynamics, Ph.D. Thesis, University of Washington, Seattle (1983).

M. Boyle, D. Lind, D. Rudolph. The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306, no. 1, (1988) 71-114.

M. Boyle and S. Tuncel, Infinite-to-one codes and Markov measures, Trans. Amer. Math. Soc. 285 (1984), 657-684.

V. Cyr, B. Kra, The automorphism group of a minimal shift of stretched exponential growth. J. Mod. Dyn. 10 (2016) 483-495.

F. Durand, B. Host, C. Skau, Substitutional dynamical systems, Bratteli diagrams and Dimension groups, Ergodic Th. & Dyn. Dyd. 19 (1999), 953-993.

D. Fiebig and U. Fiebig, Covers for coded systems, Contemporary Mathematics, 135, 1992, 139-179.

R. H. Herman, I. F. Putnuam and C. F. Skau, Ordered bratteli diagrams, Dimension groups and topological dynamics, Int. J. Math., 3 (1992), 827-864.

U. Jung, On the existence of open and bi-continuing codes, Trans. Amer. Math. Soc. 363 (2011), 1399-1417.

U. Jung, Open maps between shift spaces, Erg. Th. & Dynam. Sys. 29 (2009), 1257-1272.

P. Kurka, Topological and symbolic dynamics, Societe Mathematique de France, 2003.

D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambridge Univ. Press, 1995.

P. Walters, An introduction to ergodic theory, Springer-Verlag, 1982.


School of Mathematics,
IPM - Institute for Research in Fundamental Sciences
Niavaran Building, Niavaran Square, Tehran, Iran
Tel: +98 21 222 90 928
 Email: gt@ipm.ir