Remote access

How to Order

For AMS eBook frontlist subscriptions or backfile collection purchases:

   1a. To purchase any ebook backfile or to subscibe to the current year of Contemporary Mathematics, please download this required license agreement,

   1b. To subscribe to the current year of Memoirs of the AMS, please download this required license agreement.

   2. Complete and sign the license agreement.

   3. Email, fax, or send via postal mail to:

Customer Services
American Mathematical Society
201 Charles Street Providence, RI 02904-2294  USA
Phone: 1-800-321-4AMS (4267)
Fax: 1-401-455-4046

Visit the AMS Bookstore for individual volume purchases.

Browse the current eBook Collections price list

Powered by MathJax

Zeta functions for two-dimensional shifts of finite type

About this Title

Jung-Chao Ban, Department of Applied Mathematics, National Dong Hwa University, Hualien 97401, Taiwan., Wen-Guei Hu, Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan., Song-Sun Lin, Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan. and Yin-Heng Lin, Department of Mathematics, National Central University, ChungLi 32054, Taiwan.

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 221, Number 1037
ISBNs: 978-0-8218-7290-1 (print); 978-0-8218-9457-6 (online)
Published electronically: March 23, 2012
Keywords:Zeta functions, shift of finite type, Ising model
MSC: Primary 37B50, 37B10, 37C30; Secondary 82B20, 11M41

View full volume PDF

View other years and numbers:

Table of Contents


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


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 $\zeta =\underset {n=1}{\overset {\infty }{\prod }} \left (\det \left (I-s^{n}\tau _{n}\right )\right )^{-1}$ 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.

References [Enhancements On Off] (What's this?)