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Homoclinic points of algebraic $\mathbb Z^d$-actions


Authors: Douglas Lind and Klaus Schmidt
Journal: J. Amer. Math. Soc. 12 (1999), 953-980
MSC (1991): Primary 22D40, 54H20, 58F15; Secondary 13C10, 43A75
DOI: https://doi.org/10.1090/S0894-0347-99-00306-9
Published electronically: May 24, 1999
MathSciNet review: 1678035
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Abstract: Let $\alpha$ be an action of $\mathbb Z^d$ by continuous automorphisms of a compact abelian group $X$. A point $x$ in $X$ is called homoclinic for $\alpha$ if $\alpha^{\mathbf n}x\to 0_X$ as $\|\mathbf n\|\to\infty$. We study the set $\Delta _{\alpha}(X)$ of homoclinic points for $\alpha$, which is a subgroup of $X$. If $\alpha$ is expansive, then $\Delta _{\alpha}(X)$ is at most countable. Our main results are that if $\alpha$ is expansive, then (1) $\Delta _{\alpha}(x)$ is nontrivial if and only if $\alpha$ has positive entropy and (2) $\Delta _{\alpha}(X)$ is nontrivial and dense in $X$ if and only if $\alpha$ has completely positive entropy. In many important cases $\Delta _{\alpha}(X)$ is generated by a fundamental homoclinic point which can be computed explicitly using Fourier analysis. Homoclinic points for expansive actions must decay to zero exponentially fast, and we use this to establish strong specification properties for such actions. This provides an extensive class of examples of $\mathbb Z^d$-actions to which Ruelle's thermodynamic formalism applies. The paper concludes with a series of examples which highlight the crucial role of expansiveness in our main results.


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Additional Information

Douglas Lind
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195–4350
Email: lind@math.washington.edu

Klaus Schmidt
Affiliation: Mathematics Institute, University of Vienna, Strudlhofgasse 4, A-1090 Vienna, Austria and Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Vienna, Austria
Email: klaus.schmidt@univie.ac.at

DOI: https://doi.org/10.1090/S0894-0347-99-00306-9
Keywords: Homoclinic point, algebraic action, expansive action, specification
Received by editor(s): February 13, 1997
Received by editor(s) in revised form: May 30, 1998
Published electronically: May 24, 1999
Additional Notes: Both authors were supported in part by NSF Grant DMS-9303240. The first author was also supported in part by NSF Grant DMS-9622866.
Article copyright: © Copyright 1999 American Mathematical Society

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