ZETA FUNCTIONS FOR HIGHER-DIMENSIONAL SHIFTS OF FINITE TYPE

Abstract

This work investigates zeta functions for d-dimensional shifts of finite type, d >= 3. First, the three-dimensional case is studied. The trace operator T(a1,a2;b12) and rotational matrices R(x;a1,a2;b12) and R(y;a1,a2;b12) are introduced to study [GRAPHICS] -periodic patterns. The rotational symmetry of T(a1,a2;b12) induces the reduced trace operator tau(a1,a2;b12) and then the associated zeta function zeta(a1,a2;b12) = (det(I - s(a1a2)tau(a1,a2;b12)))(-1). The zeta function zeta is then expressed as zeta = Pi(infinity)(a1=1) Pi(infinity)(a2=1) Pi(a1-1)(b12=0) zeta(a1,a2;b12), a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL(3)(Z). Hence, a family of zeta functions exists with the same integer coefficients in their Taylor series expansions at the origin, and yields a family of identities in number theory. The methods used herein are also valid for d-dimensional cases, d >= 4, and can be applied to thermodynamic zeta functions for the three-dimensional Ising model with finite range interactions.