Zeta Functions for Multi-dimensional Shifts of Finite Type

• Main Authors: Hu, Wen-Guei, 胡文貴
• Other Authors: Lin, Song-Sun
• Format: Others
• Language: en_US
• Online Access: http://ndltd.ncl.edu.tw/handle/30520907033761886216

博士 === 國立交通大學 === 應用數學系所 === 97 === This dissertation investigates zeta functions for d-dimensional shifts of finite type, . A d-dimensional zeta function which generalizes the Artin-Mazur zeta function was given by Lind for action . First, the two-dimensional case is studied. The trace operator which is the transition matrix for x-periodic patterns of period n with height 2 is rotationally symmetric. The rotational symmetry of induces the reduced trace operator . The zeta function is now a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by 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 function is studied. The Taylor series expansions 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 methods used herein are also valid for d-dimensional cases, , and can be applied to thermodynamic zeta functions for the Ising model with finite range interactions.