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Zeta Functions for Two-Dimensional Shifts of Finite Type
Jungchao Ban, National Dong Hwa University, Hualien, Taiwan, Wen-Guei Hu and Song-Sun Lin, National Chiao Tung University, Hsinchu, Taiwan, and Yin-Heng Lin, National Central University, ChungLi, Taiwan
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Memoirs of the American Mathematical Society
2013; 60 pp; softcover
Volume: 221
ISBN-10: 0-8218-7290-7
ISBN-13: 978-0-8218-7290-1
List Price: US$60 Individual Members: US$36
Institutional Members: US\$48
Order Code: MEMO/221/1037

This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function $$\zeta^{0}(s)$$, which generalizes the Artin-Mazur zeta function, was given by Lind for $$\mathbb{Z}^{2}$$-action $$\phi$$. In this paper, the $$n$$th-order zeta function $$\zeta_{n}$$ of $$\phi$$ on $$\mathbb{Z}_{n\times \infty}$$, $$n\geq 1$$, is studied first. The trace operator $$\mathbf{T}_{n}$$, which is the transition matrix for $$x$$-periodic patterns with period $$n$$ and height $$2$$, is rotationally symmetric. The rotational symmetry of $$\mathbf{T}_{n}$$ induces the reduced trace operator $$\tau_{n}$$ and $$\zeta_{n}=\left(\det\left(I-s^{n}\tau_{n}\right)\right)^{-1}$$.

The zeta function $$\zeta=\prod_{n=1}^{\infty} \left(\det\left(I-s^{n}\tau_{n}\right)\right)^{-1}$$ 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 $$GL_{2}(\mathbb{Z})$$. Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function $$\zeta^{0}(s)$$. 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.

• Rationality of $$\zeta_{n}$$
• Equations on $$\mathbb{Z}^{2}$$ with numbers in a finite field