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Zeta Functions for Two-Dimensional Shifts of Finite Type
 
Jung-Chao Ban National Dong Hwa University, Hualien, Taiwan
Wen-Guei Hu National Chiao Tung University, Hsinchu, Taiwan
Song-Sun Lin National Chiao Tung University, Hsinchu, Taiwan
Yin-Heng Lin National Central University, ChungLi, Taiwan
Front Cover for Zeta Functions for Two-Dimensional Shifts of Finite Type
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Electronic ISBN: 978-0-8218-9457-6
Product Code: MEMO/221/1037.E
List Price: $60.00
MAA Member Price: $54.00
AMS Member Price: $36.00
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Zeta Functions for Two-Dimensional Shifts of Finite Type
Jung-Chao Ban National Dong Hwa University, Hualien, Taiwan
Wen-Guei Hu National Chiao Tung University, Hsinchu, Taiwan
Song-Sun Lin National Chiao Tung University, Hsinchu, Taiwan
Yin-Heng Lin National Central University, ChungLi, Taiwan
Available Formats:
Electronic ISBN:  978-0-8218-9457-6
Product Code:  MEMO/221/1037.E
List Price: $60.00
MAA Member Price: $54.00
AMS Member Price: $36.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 2212013; 60 pp
    MSC: Primary 37; Secondary 82; 11;

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Periodic patterns
    • 3. Rationality of $\zeta _{n}$
    • 4. More symbols on larger lattice
    • 5. Zeta functions presented in skew coordinates
    • 6. Analyticity and meromorphic extensions of zeta functions
    • 7. Equations on $\mathbb {Z}^{2}$ with numbers in a finite field
    • 8. Square lattice Ising model with finite range interaction
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Volume: 2212013; 60 pp
MSC: Primary 37; Secondary 82; 11;

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.

  • Chapters
  • 1. Introduction
  • 2. Periodic patterns
  • 3. Rationality of $\zeta _{n}$
  • 4. More symbols on larger lattice
  • 5. Zeta functions presented in skew coordinates
  • 6. Analyticity and meromorphic extensions of zeta functions
  • 7. Equations on $\mathbb {Z}^{2}$ with numbers in a finite field
  • 8. Square lattice Ising model with finite range interaction
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