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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 2012: 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 (2010): Primary 37B50, 37B10, 37C30; Secondary 82B20, 11M41

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Periodic patterns
  • Chapter 3. Rationality of
  • 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 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 , which generalizes the Artin-Mazur zeta function, was given by Lind for -action . In this paper, the th-order zeta function of on , , is studied first. The trace operator , which is the transition matrix for -periodic patterns with period and height , is rotationally symmetric. The rotational symmetry of induces the reduced trace operator and . The zeta function in the -direction is now a reciprocal of an infinite product of polynomials. The zeta function can be presented in the -direction and in the coordinates of any unimodular transformation in . Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function . 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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