Abstract
Symbolic dynamics and coding theory refer to math models of a wide variety of physical and mathematical dynamical systems, using shift spaces and the shift map σ. The technique of using symbols to code orbits goes back to the late 1800s to Hadamard, but Hedlund and Morse formalized the subject in the 1930s and 1940s, by using infinite sequences of a finite alphabet to code orbits of geodesic flows. Shannon, Weiner, and von Neumann also developed the field, independently, for more applied purposes, namely for encoding information for communication through channels.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
H. Bruin, J. Hawkins, Rigidity of smooth one-sided Bernoulli endomorphisms. New York J. Math. 15, 451–483 (2009)
T. De la Rue, 2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension. Bull. Braz. Math. Soc. 37(4), 503–521 (2006)
B. Kitchens, Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts. Universitext (Springer, Berlin, 1998)
F. Ledrappier, Un champ markovien peut être d’entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B 287(7), A561–A563 (1978)
M. Lin, Mixing for Markov operators. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 19, 231–242 (1971)
D. Lind, Multi-dimensional symbolic dynamics, in Symbolic Dynamics and Its Applications. Proceedings of Symposia in Applied Mathematics. American Mathematical Society Short Course Lecture Notes, vol. 60 (American Mathematical Society, Providence, 2004), pp. 61–79
D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding (Cambridge University Press, Cambridge, 1995)
D. Ornstein, Ergodic Theory, Randomness and Dynamical Systems (Yale University Press, New Haven, 1973)
W. Parry, P. Walters, Endomorphisms of a Lebesgue space. Bull. Amer. Math. Soc. 78, 272–276 (1972)
K. Schmidt, Algebraic Ideas in Ergodic Theory. CBMS Regional Conference Series in Mathematics, vol. 76 (American Mathematical Society, New York, 1990)
C. Shannon, W. Weaver, The Mathematical Theory of Communication (The University of Illinois Press, Urbana, 1949)
M. Taylor, Measure Theory and Integration. Graduate Studies in Mathematics, vol. 76 (American Mathematical Society, New York, 2006)
P. Walters, Roots of n:1 measure-preserving transformations. J. Lon. Math. Soc. 44, 7–14 (1969)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hawkins, J. (2021). Shift Spaces. In: Ergodic Dynamics. Graduate Texts in Mathematics, vol 289. Springer, Cham. https://doi.org/10.1007/978-3-030-59242-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-030-59242-4_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-59241-7
Online ISBN: 978-3-030-59242-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)