Zeta Functions for Two-Dimensional Shifts of Finite Type

ISBN-13: 9780821872901 / Angielski

Jungchao Ban;Wen-Guei Hu;Song-Sun Lin

Opis

This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function $zeta^(s)$, which generalizes the Artin-Mazur zeta function, was given by Lind for $mathbb^$-action $phi$. In this paper, the $n$th-order zeta function $zeta_$ of $phi$ on $mathbb_{n imes infty}$, $ngeq 1$, is studied first. The trace operator $mathbf_$, which is the transition matrix for $x$-periodic patterns with period $n$ and height $2$, is rotationally symmetric. The rotational symmetry of $mathbf_$ induces the reduced trace operator $ au_$ and $zeta_=left(detleft(I-s^ au_ ight) ight)^{-1}$. The zeta function $zeta=prod_{n=1}^ left(detleft(I-s^ au_ ight) ight)^{-1}$ 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 $GL_(mathbb)$. Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function $zeta^(s)$. 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.