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Homoclinic points, atoral polynomials, and periodic points of algebraic $\mathbb {Z}^d$-actions

Published online by Cambridge University Press:  16 May 2012

DOUGLAS LIND ,
KLAUS SCHMIDT  and
EVGENY VERBITSKIY
DOUGLAS LIND
Affiliation:
Department of Mathematics, University of Washington, Seattle, WA 98195, USA (email: [email protected])
KLAUS SCHMIDT
Affiliation:
Mathematics Institute, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria (email: [email protected])
EVGENY VERBITSKIY
Affiliation:
Mathematical Institute, University of Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, PO Box 407, 9700 AK Groningen, The Netherlands (email: [email protected])
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Abstract

Cyclic algebraic ${\mathbb {Z}^{d}}$-actions are defined by ideals of Laurent polynomials in $d$ commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative $d$-torus. For such expansive actions it is known that the limit of the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the $d$-torus in a finite set. Here we further extend it to the case where the dimension of intersection of the variety with the $d$-torus is at most $d-2$. The main tool is the construction of homoclinic points which decay rapidly enough to be summable.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 4 , August 2013 , pp. 1060 - 1081
Copyright
Copyright © 2012 Cambridge University Press 

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