Homoclinic points, atoral polynomials, and periodic points of algebraic Z(d)-actions

Douglas Lind*, Klaus Schmidt, Evgeny Verbitskiy

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

13 Citations (Scopus)


Cyclic algebraic 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.

Original languageEnglish
Pages (from-to)1060-1081
Number of pages22
JournalErgodic Theory and Dynamical Systems
Issue number4
Publication statusPublished - Aug-2013
Externally publishedYes



Cite this