Zeta functions for two-dimensional shifts of finite type

Jung-Chao Ban National Hualien University of Education Wen-Guei Hu National Chiao Tung University Song-Sun Lin National Chiao Tung University Yin-Heng Lin National Central University

Dynamical Systems mathscidoc:1609.11001

Memoirs of the American Mathematical Society, 221, (1037), 2013
This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function ζ0(s) which generalizes the Artin-Mazur zeta function was given by Lind for Z2-action φ. The n-th order zeta function ζn of φ on Zn×∞, n ≥ 1, is studied first. The trace operator Tn which is the transition matrix for x-periodic patterns of period n with height 2 is rotationally symmetric. The rotational symmetry of Tn induces the reduced trace operator τn and ζn = (det (I − snτn))−1. The zeta function ζ =∞ Qn=1 (det (I − snτn))−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 GL2(Z). Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function ζ0(s). The Taylor series at the origin for these zeta functions 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.
Zeta function; shift of finite type; trace operator; periodic pattern
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@inproceedings{jung-chao2013zeta,
  title={Zeta functions for two-dimensional shifts of finite type},
  author={Jung-Chao Ban, Wen-Guei Hu, Song-Sun Lin, and Yin-Heng Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914154813067199003},
  booktitle={Memoirs of the American Mathematical Society},
  volume={221},
  number={1037},
  year={2013},
}
Jung-Chao Ban, Wen-Guei Hu, Song-Sun Lin, and Yin-Heng Lin. Zeta functions for two-dimensional shifts of finite type. 2013. Vol. 221. In Memoirs of the American Mathematical Society. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914154813067199003.
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