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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Expansive Subdynamics

Authors: Mike Boyle and Douglas Lind
Journal: Trans. Amer. Math. Soc. 349 (1997), 55-102
MSC (1991): Primary 54H20, 58F03; Secondary 28D20, 28D15, 28F15, 58F11, 58F08
MathSciNet review: 1355295
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Abstract: This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let $\alpha $ be a continuous action of ${\mathbb Z}^d $ on an infinite compact metric space. For each subspace $V$ of ${\mathbb R}^d $ we introduce a notion of expansiveness for $\alpha $ along $V$, and show that there are nonexpansive subspaces in every dimension $\le d-1$. For each $k\le d$ the set ${\mathbb E} _k(\alpha )$ of expansive $k$-dimensional subspaces is open in the Grassmann manifold of all $k$-dimensional subspaces of ${\mathbb R}^d $. Various dynamical properties of $\alpha $ are constant, or vary nicely, within a connected component of ${\mathbb E} _k(\alpha )$, but change abruptly when passing from one expansive component to another. We give several examples of this sort of ``phase transition,'' including the topological and measure-theoretic directional entropies studied by Milnor, zeta functions, and dimension groups. For $d=2$ we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an ${\mathbb E} _1(\alpha )$. The unresolved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our analysis.

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Additional Information

Mike Boyle
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Douglas Lind
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195–4350

Keywords: Expansive, subdynamics, symbolic dynamics, entropy, directional entropy, shift of finite type, group automorphism.
Received by editor(s): May 6, 1994
Additional Notes: The first author was supported in part by NSF Grants DMS-8802593, DMS-9104134, and DMS-9401538.
The second author was supported in part by NSF Grants DMS-9004253 and DMS-9303240.
Article copyright: © Copyright 1997 American Mathematical Society