Electronic ISBN:  9780821894576 
Product Code:  MEMO/221/1037.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 221; 2013; 60 ppMSC: Primary 37; Secondary 82; 11;
This work is concerned with zeta functions of twodimensional shifts of finite type. A twodimensional zeta function \(\zeta^{0}(s)\), which generalizes the ArtinMazur zeta function, was given by Lind for \(\mathbb{Z}^{2}\)action \(\phi\). In this paper, the \(n\)thorder 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 \(\zeta_{n}=\left(\det\left(Is^{n}\tau_{n}\right)\right)^{1}\).
The zeta function \(\zeta=\prod_{n=1}^{\infty} \left(\det\left(Is^{n}\tau_{n}\right)\right)^{1}\) 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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This work is concerned with zeta functions of twodimensional shifts of finite type. A twodimensional zeta function \(\zeta^{0}(s)\), which generalizes the ArtinMazur zeta function, was given by Lind for \(\mathbb{Z}^{2}\)action \(\phi\). In this paper, the \(n\)thorder 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 \(\zeta_{n}=\left(\det\left(Is^{n}\tau_{n}\right)\right)^{1}\).
The zeta function \(\zeta=\prod_{n=1}^{\infty} \left(\det\left(Is^{n}\tau_{n}\right)\right)^{1}\) 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