Zeta Functions for Higher-Dimensional Shifts of Finite Type

This work investigates zeta functions for d-dimensional shifts of finite type, d ≥ 3. First, the three-dimensional case is studied. The trace operator Ta1,a2;b12 and rotational matrices Rx;a1,a2;b12 and Ry;a1,a2;b12 are introduced to study ${\scriptsize\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{}} a_{1} & b_{12} & b_{23}\\[1pt] 0 & a_{2} & b_{23} \\[1pt] 0 & 0 & a_{3} \end{array}\right]}$ -periodic patterns. The rotational symmetry of Ta1,a2;b12 induces the reduced trace operator τa1,a2;b12 and then the associated zeta function ζa1,a2;b12 = (det(I-sa1a2τa1,a2;b12))-1. The zeta function ζ is then expressed as $\zeta=\prod_{a_{1}=1}^{\infty}\prod_{a_{2}=1}^{\infty} \prod_{b_{12}=0}^{a_{1}-1}\zeta_{a_{1},a_{2};b_{12}}$, a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL3(ℤ). Hence, a family of zeta functions exists with the same integer coefficients in their Taylor series expansions at the origin, and yields a fami...