### Abstract

This work is concerned with zeta functions of two-dimensional shifts of finite type. A two-dimensional zeta function ζ^{0}(s), which generalizes the Artin-Mazur zeta function, was given by Lind for ℤ^{2}-action φ. In this paper, the nth-order zeta function ζ_{n} of φ on ℤ_{n×8}, n ≥ 1, is studied first. The trace operator T_{n}, which is the transition matrix for x-periodic patterns with period n and height 2, is rotationally symmetric. The rotational symmetry of Tninduces the reduced trace operator τ_{n} and ζ_{n}= (det(I - s^{n}τ_{n}))^{-1}. The zeta function ζ = Π ^{∞}_{n=1}(det(I - s ^{n}τ_{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}(ℤ). Therefore, there exists a family of zeta functions that are meromorphic extensions of the same analytic function ζ^{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.

Original language | English |
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Pages (from-to) | 1-62 |

Number of pages | 62 |

Journal | Memoirs of the American Mathematical Society |

Volume | 221 |

Issue number | 1037 |

DOIs | |

State | Published - 1 Jan 2013 |

### Keywords

- Ising model
- Shift of finite type
- Zeta functions

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## Cite this

*Memoirs of the American Mathematical Society*,

*221*(1037), 1-62. https://doi.org/10.1090/S0065-9266-2012-00653-8