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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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