### Abstract

This work investigates zeta functions for d-dimensional shifts of finite type, d < 3. First, the three-dimensional case is studied. The trace operator T_{a}1,a_{2};b_{12} and rotational matrices Rx;a _{1},a_{2};b_{12} and Ry;a_{1},a _{2};b_{12} are introduced to study ${\scriptsize\left[\ begin{array}{@{}c@{\quad}c@{\quad}c@{}} a-{1} & b-{12} & b-{23}\\[1pt] 0 & a-{2} & b-{23} \\[1pt] 0 & 0 & a-{3} \end{array}\right]} $ -periodic patterns. The rotational symmetry of T_{a}1,a _{2};b_{12}induces the reduced trace operator τa _{1},a_{2};b_{12} and then the associated zeta function ζ_{a}1,a_{2};b_{12} = (det(I-sa_{1}a _{2}τ_{a}1,a_{2};b_{12}))^{-1}. The zeta function ζ is then expressed as ζ=∏ _{a1=1}∏_{a2=1} ∏_{b12=0} ^{a1-1}ζa1,a2;b12, reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL_{3}(). Hence, a family of zeta functions exists with the same integer coefficients in their Taylor series expansions at the origin, and yields a family of identities in number theory. The methods used herein are also valid for d-dimensional cases, d < 4, and can be applied to thermodynamic zeta functions for the three-dimensional Ising model with finite range interactions.

Original language | English |
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Pages (from-to) | 3671-3689 |

Number of pages | 19 |

Journal | International Journal of Bifurcation and Chaos |

Volume | 19 |

Issue number | 11 |

DOIs | |

State | Published - 1 Jan 2009 |

### Keywords

- Cellular neural networks
- Ising model
- Patterns generation problem
- Phase-transition
- Shift of finite type
- Zeta function

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## Cite this

*International Journal of Bifurcation and Chaos*,

*19*(11), 3671-3689. https://doi.org/10.1142/S0218127409025055