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

TL;DR

This paper extends the understanding of periodic points and entropy in algebraic Z^d-actions by analyzing cases where the associated variety intersects the d-torus in higher dimensions, using homoclinic points.

Contribution

It generalizes previous results on the growth rate of periodic points to cases with higher-dimensional intersections, employing homoclinic point constructions.

Findings

Established the existence of the limit for the growth rate of periodic points in new intersection cases.

Extended the class of algebraic actions for which entropy and periodic point growth are well-understood.

Developed methods to construct summable homoclinic points for these actions.

Abstract

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 for 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 when 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.