Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian ${\mathbb{Z}}^d$-actions

Abstract

For every $P\in \mathbb{Z}[x_1^{\pm 1},\dots ,x_d^{\pm 1}]\setminus \{0\}$, and every $\epsilon >0$, we prove that there are a computable function $M=M(d,\epsilon ,°P,h(P))<\infty$ and a finite union $Z=Z(d,\epsilon ,°P,h(P))$ of proper torsion cosets $\mathbf{\mu }T⊊{\mathbb{G}}_m^d$ such that, for every $N\in \mathbb{N}$, $Z$ contains all but at most $M$ of the torsion points $\mathbf{\zeta }\in {\mu }_N^d$ satisfying $|P(\mathbf{\zeta })|<e^{-\epsilon \varphi (N)}$. This extends a well known structural theorem from torsion points lying exactly on a variety to torsion points lying very near to the subvariety. As a consequence, we prove that the averages of $\log |P(\mathbf{x})|$ over ${\mu }_N^d$ converge as $N\to \infty$ to the Mahler measure of $P$. By the work of B. Kitchens, D. Lind, K. Schmidt and T. Ward, the convergence consequence amounts to the following statement in dynamics: For every Noetherian ${\mathbb{Z}}^d$-action $T:{\mathbb{Z}}^d\to \mathrm{Aut}(X)$ by automorphisms of a compact abelian group $X$ having a finite topological entropy $h(T)$, the exponential growth rate of the number of connected components of the group ${\mathrm{Per}}_N(T)$ of $N\cdot {\mathbb{Z}}^d$-periodic points of $(X,T)$ exists as $N\to \infty$, and equals the topological entropy $h(T)$. Moreover, it follows that all weak-$\ast$ limit measures of the push-forwards of the Haar measures on ${\mathrm{Per}}_N(T)$, under any a sequence of positive integers $N$, are measures of maximum entropy $h(T)$. Our main arithmetic result extends to Diophantine approximation by points of sufficiently small canonical height. It is best possible in such a generality, where an exceptional set is an inevitable feature.

39 pages; added a section on the growth of homology in abelian congruence covers

References in corpus (1)

• Diophantine Approximations on Definable Sets

Cited by in corpus (2)

• Graph complexity and Mahler measure
• Graph Complexity and Link Colorings