Entropy and growth rate of periodic points of algebraic Z^d-actions

TL;DR

This paper extends the understanding of the growth rate of periodic points in algebraic Z^d-actions, showing that the limit exists and equals entropy even when the complex variety intersects the unit torus in finitely many points.

Contribution

It generalizes previous results by establishing the growth rate and entropy equality for a broader class of algebraic actions with finite intersections.

Findings

The growth rate of periodic points exists for these actions.

The growth rate equals the entropy of the action.

Homoclinic points are used as a key technical tool.

Abstract

Expansive algebraic Z^d-actions corresponding to ideals are characterized by the property that the complex variety of the ideal is disjoint from the multiplicative unit torus. For such actions it is known that the limit for the growth rate of periodic points exists and equals the entropy of the action. We extend this result to actions for which the complex variety intersects the multiplicative torus in a finite set. The main technical tool is the use of homoclinic points which decay rapidly enough to be summable.