 Ban, JungChao, 1974
 Providence, Rhode Island : American Mathematical Society, 2013.
 Description
 Book — v, 60 pages ; 26 cm.
 Summary

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.
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