Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian ℤ^d-actions

2016/11/15 by Vesselin Dimitrov, Dimitrov, Vesselin
Computer Science · Mathematics · Physics and Astronomy · #11G50 #11J17 #11J25 #11J87 #11K60 #11R06 #37A25 #37A35 #37A45 #57M25 #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Theoretical and Computational Physics #Topological and Geometric Data Analysis #math.DS #math.NT #msc:11G50 #msc:11J17 #msc:11J25 #msc:11J87 #msc:11K60 #msc:11R06 #msc:37A25 #msc:37A35 #msc:37A45 #msc:57M25

paper · pdf · doi:10.48550/arxiv.1611.04664

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

openalex publication_date 2016/11/15 · arxiv created 2017/01/01 · arxiv updated 2017/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For every P ∈ ℤ[x₁^{± 1}, …, x_d^{± 1}] ∖ \0\, and every ε > 0, we prove that there are a computable function M = M(d,ε,°P,h(P)) < ∞ and a finite union Z = Z(d,ε,°P,h(P)) of proper torsion cosets \boldsymbolμ T \subsetneq \mathbbG_m^d such that, for every N ∈ ℕ, Z contains all but at most M of the torsion points \boldsymbolζ ∈ μ_N^d satisfying |P(\boldsymbolζ)| < e^{-ε ϕ(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(x)| over μ_N^d converge as N → ∞ 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 ℤ^d-action T : ℤ^d → 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 Per_N(T) of N ⋅ ℤ^d-periodic points of (X,T) exists as N → ∞, and equals the topological entropy h(T). Moreover, it follows that all weak-* limit measures of the push-forwards of the Haar measures on 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.

Citations

• A lower bound on the orbit growth of a regular self-map of affine space
• Mahler measure and entropy for commuting automorphisms of compact groups
• The structure of skew products with ergodic group automorphisms
• Homoclinic points, atoral polynomials, and periodic points of algebraic -actions

Related

• Mahler measure, links and homology growth
• Homology torsion growth and Mahler measure
• Mahler's Work and Algebraic Dynamical Systems
• The asymptotic Mahler measure of Gaussian periods
• Wandering points for the Mahler measure