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Non-expansive directions for ℤ2 actions

Published online by Cambridge University Press:  24 March 2010

MICHAEL HOCHMAN
MICHAEL HOCHMAN*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA (email: [email protected])
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Abstract

We show that any direction in the plane occurs as the unique non-expansive direction of a ℤ2 action, answering a question of Boyle and Lind. In the case of rational directions, the subaction obtained is non-trivial. We also establish that a cellular automaton acting on a subshift can have zero Lyapunov exponents and at the same time act sensitively; and, more generally, for any positive real θ there is a cellular automaton acting on an appropriate subshift with λ+=−λ−=θ.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 1 , February 2011 , pp. 91 - 112
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Albert, J. and Culik, K. II. A simple universal cellular automaton and its one-way and totalistic version. Complex Systems 1(1) (1987), 1–16.Google Scholar
[2]Boyle, M.. Open problems in symbolic dynamics. Geometric and Probabilistic Structures in Dynamics (Contemporary Mathematics, 469). American Mathematical Society, Providence, RI, 2008, pp. 69–118.CrossRefGoogle Scholar
[3]Boyle, M. and Lind, D.. Expansive subdynamics. Trans. Amer. Math. Soc. 349(1) (1997), 55–102.CrossRefGoogle Scholar
[4]Bressaud, X. and Tisseur, P.. On a zero speed sensitive cellular automaton. Nonlinearity 20(1) (2007), 1–19.CrossRefGoogle Scholar
[5]Gács, P.. Reliable cellular automata with self-organization. J. Stat. Phys. 103(1–2) (2001), 45–267.CrossRefGoogle Scholar
[6]Madden, K. M.. A single nonexpansive, nonperiodic rational direction. Complex Systems 12(2) (2000), 253–260.Google Scholar
[7]Morita, K. and Harao, M.. Computation universality of one-dimensional reversible (injective) cellular automata. Trans. Inst. Electron. Inf. Commun. Eng. E72 (1989), 758–762.Google Scholar
[8]Shereshevsky, M. A.. Lyapunov exponents for one-dimensional cellular automata. J. Nonlinear Sci. 2(1) (1992), 1–8.CrossRefGoogle Scholar
[9]Tisseur, P.. Cellular automata and Lyapunov exponents. Nonlinearity 13(5) (2000), 1547–1560.CrossRefGoogle Scholar
[10]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, New York, 1982.CrossRefGoogle Scholar