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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 Click above image for expanded view 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.

• 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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