Homoclinic points of algebraic actions
Authors:
Douglas Lind and Klaus Schmidt
Journal:
J. Amer. Math. Soc. 12 (1999), 953980
MSC (1991):
Primary 22D40, 54H20, 58F15; Secondary 13C10, 43A75
Published electronically:
May 24, 1999
MathSciNet review:
1678035
Fulltext PDF Free Access
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Abstract: Let be an action of by continuous automorphisms of a compact abelian group . A point in is called homoclinic for if as . We study the set of homoclinic points for , which is a subgroup of . If is expansive, then is at most countable. Our main results are that if is expansive, then (1) is nontrivial if and only if has positive entropy and (2) is nontrivial and dense in if and only if has completely positive entropy. In many important cases 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 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, A1090 Vienna, Austria and Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A1090 Vienna, Austria
Email:
klaus.schmidt@univie.ac.at
DOI:
http://dx.doi.org/10.1090/S0894034799003069
PII:
S 08940347(99)003069
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 DMS9303240. The first author was also supported in part by NSF Grant DMS9622866.
Article copyright:
© Copyright 1999
American Mathematical Society
