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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)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00653-8
Published electronically: March 23, 2012
Keywords:Zeta functions, shift of finite type, Ising model
MSC (2010): Primary 37B50, 37B10, 37C30; Secondary 82B20, 11M41

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Table of Contents


Chapters

  • 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

Abstract


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