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

DOI:
https://doi.org/10.1090/S0002-9947-97-01634-6

MathSciNet review:
1355295

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Abstract: This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let be a continuous action of on an infinite compact metric space. For each subspace of we introduce a notion of expansiveness for along , and show that there are nonexpansive subspaces in every dimension . For each the set of expansive -dimensional subspaces is open in the Grassmann manifold of all -dimensional subspaces of . Various dynamical properties of are constant, or vary nicely, within a connected component of , 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 we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an . 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

Email:
mmb@math.umd.edu

**Douglas Lind**

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

Email:
lind@math.washington.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01634-6

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