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Symbolic Dynamics working group

The seminar will meet on Wednesdays, at 12:30, in MW 154. A description of the topic of the seminar is below. For more information please contact Vitaly Bergelson (vitaly@math.ohio-state.edu).

Pending talks:

  • February 10, Kostyantyn Medynets, Topological Entropy: Definition and Examples

    Abstract: We will give the definition of an entropy for a topological dynamical system. We will find exact value of the entropy for some symbolic dynamical systems such as Markov chains, shifts of finite type, etc.

Past talks:

  • February 3, Exercises and open problems in symbolic dynamics
  • January 27, Yuri Gomes Lima, Symbolic dynamics with prescribed topological and ergodic properties - II
  • January 20, Yuri Gomes Lima, Symbolic dynamics with prescribed topological and ergodic properties

    Abstract: We discuss a construction due to Hahn and Katznelson to obtain symbolic dynamical systems with prescribed topological and ergodic properties, such as minimality, unique ergodicity and positive entropy. If the time permits, we also discuss a modification of it to get a negative answer to a question raised by V. Bergelson.

  • January 13, Charles Baker, Bi-Infinite Symbolic Dynamics and the Smale Horseshoe

    Abstract: We review the Smale Horseshoe map and explain why its stable set is topologically conjugate to the full bi-infinite shift on 2 elements. We also will explain how this equivalence to a full shift relates to the notion of chaos in the map [and, if time permits, Hausdorff dimension].

  • December 2, Marc Carnovale, Geometric Constructions and Thermodynamic Formalism

    Abstract: Given a finite collection l_1, ... , l_p of numbers in (0,1), with sum less than 1, choose p arbitrary non-overlappiing subintervals J_1, ..., J_p of (0,1) of lengths l_1, ..., l_p. In each J_i, choose p nonoverlapping subintervals J_i1, ..., J_ip of lengths l_i*l_j, j = 1, ..., p. Repeat the procedure inductively. The limit set is the intersection over all natural numbers n of the collection of all of the intervals J_ij...n chosen at the nth step. With the aid of symbolic dynamics and the thermodynamic formalism, Pesin and Weiss generalized a result of Moran stating that the Hausdorff dimension of the limit set is a constant dependent only on the numbers l_1, ..., l_p, irrespective of how the intervals are chosen. We review the thermodynamic formalism and the proof of this fact, and consider a generalization of this result due to Barreira involving the non-additive topological pressure.

  • November 18, Kostyantyn Medynets, Unique ergodicity of minimal symbolic flows

    Abstract: We will review the paper by M. Boshernitzan "A condition for unique ergodicity of minimal symbolic flows", which gives a sufficient condition for unique ergodicity in terms of the "block growth speed". Furthermore, in some situations the speed of growth provides a bound on the number of ergodic invariant measures.

  • November 4, David Ralston, Counting Factors

    Abstract: We will introduce the "block-growth function," which counts the number of factors of different lengths which appear in an infinite word. The rate at which this monotone function grows encodes much information about the original word. We will investigate the minimal non-trivial block growth rate, explicit examples of linear and other polynomial rates, and ask questions about other possible growth rates.

  • October 28, Jim Tseng, Symbolic dynamics, nondense orbits, and winning sets (Part 2)

    Abstract: I will continue to talk about nondense orbits. I will give a brief outline of the proof that the set of points with nondense orbits under the multiplication-by-2 map on the circle has Hausdorff dimension 1. I will define winning sets (which is a strengthening of the notion of full Hausdorff dimension) and give examples related to nondense orbits.

  • October 21, Jim Tseng, Symbolic dynamics, nondense orbits, and winning sets

    Abstract: To a smooth (enough) expanding self-map of a (smooth, connected) compact Riemannian manifold, one can associate a symbolic dynamical system via the construction of a Markov partition, which is a certain family of subsets of the manifold. This symbolic description is detailed enough to allow us to study the set of points with nondense orbits. This set is a null set but with full Hausdorff dimension. For circles, this set is also winning, which is a strengthening of the notion of full Hausdorff dimension.

    As part of the talk, I will define Markov partition, Hausdorff dimension, and winning sets.

  • October 14, Bill Mance, Many facets of normality (continuation)
  • October 7, Bill Mance, Many facets of normality
    Abstract: We will discuss the concept of a normal number as it applies to $b$-ary expansions, continued fractions, Luroth series, $\beta$ expansions and the Cantor series expansion. We discuss the connections between normality, uniform distribution and dynamical systems and show explicit examples of normal numbers for different expansions.
  • September 30, Sasha Leibman, Van der Waerden and Szemeredi theorems via symbolic dynamics
    Abstract: I will show how the combinatorial van der Waerden and Szemeredi theorems can be translated to the language of symbolic dynamics. This will be a very elementary, introductory talk, intended for novices in the field; nobody is obligated to attend.

Symbolic dynamics arises naturally in areas as diverse as applied mathematics, number theory, combinatorics, probability, ergodic theory, coding and information theory. A plethora of motivating examples and numerous connections to other fields make symbolic dynamics very attractive and suitable for students at both the undergraduate and graduate level.

The term symbolic dynamics refers to dynamical systems on spaces of sequences, with dynamics induced by the shift. These systems are readily introduced (any shift-invariant set of sequences from a fixed alphabet is a symbolical dynamical system!) and can be used to motivate, illustrate and study various key concepts in dynamical systems such as ergodicity, entropy, invariant measures and many others. They also contribute new tools to other areas. The following, by far not exhaustive, list of applications of symbolic dynamics demonstrates its ubiquity and potential:

  • Hyperbolic diffeomorphisms, geodesic flows and billiards [2,10]
  • Data storage and transmission in coding and information theory [7]
  • Perron-Frobenius theory of nonnegative matrices [1]
  • Ramsey theory and additive number theory via ergodic theory [4]
  • Markov chains and probability [6]
  • Mathematical linguistics [8]
  • Chaos, Smale's horseshoe, Sharkovski's theorem [9]
  • Homoclinic and heteroclinic cycles in ODEs [5]
  • Applications to rotating convection rolls in Rayleigh-Benard convection and forced oscillations of elastic beams in magnetic fields [5]

References:

[1] M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics. Ann. Math. 133 (1991) 249-316.
[2] L.A. Bunimovich and Y.G. Sinai, Markov partitions for dispersed billiards. Comm. Math. Phys. 73 (1980) 247-280.
[3] K. Dajani and C. Kraaikamp, Ergodic theory of numbers. Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, (2002)
[4] H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory. Princeton University Press, 1981.
[5] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York, 1983.
[6] B. Kitchens, Symbolic Dynamics. Springer, 1998.
[7] D. Lind and B. Marcus, Symbolic Dynamics and Coding. Cambridge University Press, 1995.
[8] M. Lothaire, Combinatorics on Words. Cambridge University Press, 1983.
[9] J. Palis and F. Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations. Cambridge University Press, 1993.
[10] C. Series, Symbolic dynamics for geodesic flows. Acta Math. 146 (1981) 103-128.

Additional litearature:

  1. J. C. Alexander, The symbolic dynamics of the sequence of pedal triangles. Math. Mag. 66 (1993), no. 3, 147--158.
  2. J-P. Allouche, J. Shallit, The ubiquitous Prouhet-Thue-Morse sequence. Sequences and their applications (Singapore, 1998), 1-16, Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 1999.
  3. B. Aulbach and B. Kieninger, An elementary proof for hyperbolicity and chaos of the logistic maps. J. Difference Equ. Appl. 10 (2004), no. 13-15, 1243-1250.
  4. J. Banks and V. Dragan, Smale's horseshoe map via ternary numbers. (English summary) SIAM Rev. 36 (1994), no. 2, 265-271.
  5. R. M. Corless, Continued fractions and chaos. Amer. Math. Monthly 99 (1992), no. 3, 203-215.
  6. E. M. Coven and G. A. Hedlund, Sequences with minimal block growth. Math. Systems Theory 7 (1973), 138-153.
  7. G. A. Hedlund, Sturmian minimal sets. Amer. J. Math. 66, (1944), 605-620.
  8. Kraft, Roger L. Chaos, Cantor sets, and hyperbolicity for the logistic maps. Amer. Math. Monthly 106 (1999), no. 5, 400-408.
  9. M. Mendes France, Some applications of the theory of automata. Prospects of mathematical science (Tokyo, 1986), 127-140. World Sci. Publishing, Singapore, 1988.
  10. M. Morse and G. Hedlund, Symbolic dynamics, Amer. J. Math. 60 (1938), 815-866.
  11. M. Morse and G. Hedlund, Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62, (1940), 1-42.
  12. M. Morse and G. Hedlund, Unending chess, symbolic dynamics and a problem in semigroups. Duke Math. J. 11, (1944), 1-7.
  13. R. Oldenburger, Exponent trajectories in symbolic dynamics. Trans. Amer. Math. Soc. 46, (1939), 453-466.
  14. R. Oldenburger, Recurrence of symbolic elements in dynamics. Bull. Amer. Math. Soc. 47, (1941), 294-297.
  15. K. E. Petersen, A topologically strongly mixing symbolic minimal set. Trans. Amer. Math. Soc. 148 (1970), 603-612.


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Last updated by Alexander Leibman on 02/08/10