article · Transactions of the American Mathematical Society · 1997

Expansive Subdynamics

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Abstract

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a continuous action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper Z Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {Z}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on an infinite compact metric space. For each subspace <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we introduce a notion of expansiveness for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> along <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and show that there are nonexpansive subspaces in every dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="less-than-or-equal-to d minus 1"> <mml:semantics> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\le d-1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k less-than-or-equal-to d"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">k\le d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E Subscript k Baseline left-parenthesis alpha right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {E}_k(\alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of expansive <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional subspaces is open in the Grassmann manifold of all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional subspaces of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Various dynamical properties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are constant, or vary nicely, within a connected component of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E Subscript k Baseline left-parenthesis alpha right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {E}_k(\alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d equals 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E 1 left-parenthesis alpha right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>α</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {E}_1(\alpha )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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.

Cite this paper

APA 7

Boyle, M., & Lind, D. (1997). Expansive Subdynamics. Transactions of the American Mathematical Society, 349(1), 55–102. https://doi.org/10.1090/s0002-9947-97-01634-6

MLA 9

Boyle, Mike, and Douglas Lind. “Expansive Subdynamics.” Transactions of the American Mathematical Society, vol. 349, no. 1, 1997, pp. 55–102, https://doi.org/10.1090/s0002-9947-97-01634-6.

Chicago

Boyle, Mike, and Douglas Lind. 1997. “Expansive Subdynamics.” Transactions of the American Mathematical Society 349 (1): 55–102. https://doi.org/10.1090/s0002-9947-97-01634-6.

Harvard

Boyle, M. and Lind, D. (1997) ‘Expansive Subdynamics’, Transactions of the American Mathematical Society, 349(1), pp. 55–102. Available at: https://doi.org/10.1090/s0002-9947-97-01634-6.

BibTeX

@article{boyle1997expansive,
  title = {{Expansive Subdynamics}},
  author = {Boyle, Mike and Lind, Douglas},
  journal = {Transactions of the American Mathematical Society},
  volume = {349},
  number = {1},
  pages = {55--102},
  year = {1997},
  doi = {10.1090/s0002-9947-97-01634-6}
}

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