Homoclinic points, atoral polynomials, and periodic points of algebraic \(\mathbb{Z}^d\)-actions (Q2842227)

In this paper the authors extend the main result of \textit{D. Lind} et al. [Contemp. Math. 532, 195--211 (2010; Zbl 1217.54034)], that under certain conditions the growth rate of periodic points of a cyclic algebraic \(\mathbb{Z}^d\)-action on the torus exists and is precisely the entropy of the action. Specifically, it is proven that the result holds if the ideal of Laurent polynomials defining the action intersects the \(d\)-torus in an at most \((d-2)\)-dimensional set. The previous condition required a finite intersection. Notably, both results do not assume expansiveness; a number of illustrative examples are given.NEWLINENEWLINEAs in [loc. cit.], the main tool is to construct homoclinic points that are summable. However, in the present paper the authors do not rely on \textit{A. O. Gelfond}'s estimate [Transcendental and algebraic numbers. New York: Dover Publications, Inc. (1960; Zbl 0090.26103)], but employ an analogous diophantine estimate, which is equivalent to the existence of summable homoclinic points.