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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

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
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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.

Table of Contents

  • Introduction
  • Periodic patterns
  • Rationality of \(\zeta_{n}\)
  • More symbols on larger lattice
  • Zeta functions presented in skew coordinates
  • Analyticity and meromorphic extensions of zeta functions
  • Equations on \(\mathbb{Z}^{2}\) with numbers in a finite field
  • Square lattice Ising model with finite range interaction
  • Bibliography
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