# Zeta Functions for Two-Dimensional Shifts of Finite Type

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*Jungchao Ban; Wen-Guei Hu; Song-Sun Lin; Yin-Heng Lin*

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

# Table of Contents

## Zeta Functions for Two-Dimensional Shifts of Finite Type

- Chapter 1. Introduction 18 free
- Chapter 2. Periodic patterns 714 free
- Chapter 3. Rationality of 𝜁_{𝑛} 1926
- Chapter 4. More symbols on larger lattice 2734
- Chapter 5. Zeta functions presented in skew coordinates 3138
- Chapter 6. Analyticity and meromorphic extensions of zeta functions 3946
- Chapter 7. Equations on ℤ² with numbers in a finite field 4754
- Chapter 8. Square lattice Ising model with finite range interaction 5360
- Bibliography 5966