Zeta functions for higher-dimensional shifts of finite type

Wen Guei Hu*, Song-Sun Lin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

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 Ta1,a2;b12 and rotational matrices Rx;a 1,a2;b12 and Ry;a1,a 2;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,a 2;b12induces the reduced trace operator τa 1,a2;b12 and then the associated zeta function ζa1,a2;b12 = (det(I-sa1a 2τa1,a2;b12))-1. The zeta function ζ is then expressed as ζ=∏ a1=1a2=1b12=0 a1-1ζa1,a2;b12, reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL3(). 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.

Original language English 3671-3689 19 International Journal of Bifurcation and Chaos 19 11 https://doi.org/10.1142/S0218127409025055 Published - 1 Jan 2009

Keywords

• Cellular neural networks
• Ising model
• Patterns generation problem
• Phase-transition
• Shift of finite type
• Zeta function