Zeta functions for two-dimensional shifts of finite type

Abstract

This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function ζ0(s), which generalizes the Artin-Mazur zeta function, was given by Lind for ℤ2-action φ. In this paper, the nth-order zeta function ζn of φ on ℤn×8, n ≥ 1, is studied first. The trace operator Tn, which is the transition matrix for x-periodic patterns with period n and height 2, is rotationally symmetric. The rotational symmetry of Tninduces the reduced trace operator τn and ζn= (det(I - snτn))-1. The zeta function ζ = Π ∞n=1(det(I - s nτn))-1 in the x-direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the y-direction and in the coordinates of any unimodular transformation in GL2(ℤ). Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function ζ0(s). The natural boundary of zeta functions is studied. The Taylor series for these zeta functions at the origin are equal with integer coefficients, yielding a family of identities, which are of interest in number theory. The method applies to thermodynamic zeta functions for the Ising model with finite range interactions. © 2012 by the American Mathematical Society. All rights reserved.