{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T13:33:31Z","timestamp":1648992811489},"reference-count":42,"publisher":"World Scientific Pub Co Pte Lt","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2009,11]]},"abstract":"<jats:p>This work investigates zeta functions for d-dimensional shifts of finite type, d \u2265 3. First, the three-dimensional case is studied. The trace operator T<jats:sub>a<jats:sub>1<\/jats:sub>,a<jats:sub>2<\/jats:sub>;b<jats:sub>12<\/jats:sub><\/jats:sub>and rotational matrices R<jats:sub>x;a<jats:sub>1<\/jats:sub>,a<jats:sub>2<\/jats:sub>;b<jats:sub>12<\/jats:sub><\/jats:sub>and R<jats:sub>y;a<jats:sub>1<\/jats:sub>,a<jats:sub>2<\/jats:sub>;b<jats:sub>12<\/jats:sub><\/jats:sub>are introduced to study [Formula: see text] -periodic patterns. The rotational symmetry of T<jats:sub>a<jats:sub>1<\/jats:sub>,a<jats:sub>2<\/jats:sub>;b<jats:sub>12<\/jats:sub><\/jats:sub>induces the reduced trace operator \u03c4<jats:sub>a<jats:sub>1<\/jats:sub>,a<jats:sub>2<\/jats:sub>;b<jats:sub>12<\/jats:sub><\/jats:sub>and then the associated zeta function \u03b6<jats:sub>a<jats:sub>1<\/jats:sub>,a<jats:sub>2<\/jats:sub>;b<jats:sub>12<\/jats:sub><\/jats:sub>= ( det (I-s<jats:sup>a<jats:sub>1<\/jats:sub>a<jats:sub>2<\/jats:sub><\/jats:sup>\u03c4<jats:sub>a<jats:sub>1<\/jats:sub>,a<jats:sub>2<\/jats:sub>;b<jats:sub>12<\/jats:sub><\/jats:sub>))<jats:sup>-1<\/jats:sup>. The zeta function \u03b6 is then expressed as [Formula: see text], a reciprocal of an infinite product of polynomials. The results hold for any inclined coordinates, determined by unimodular transformation in GL<jats:sub>3<\/jats:sub>(\u2124). 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 \u2265 4, and can be applied to thermodynamic zeta functions for the three-dimensional Ising model with finite range interactions.<\/jats:p>","DOI":"10.1142\/s0218127409025055","type":"journal-article","created":{"date-parts":[[2010,2,24]],"date-time":"2010-02-24T01:07:55Z","timestamp":1266973675000},"page":"3671-3689","source":"Crossref","is-referenced-by-count":2,"title":["ZETA FUNCTIONS FOR HIGHER-DIMENSIONAL SHIFTS OF FINITE TYPE"],"prefix":"10.1142","volume":"19","author":[{"given":"WEN-GUEI","family":"HU","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu 30010, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"SONG-SUN","family":"LIN","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu 30010, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.2307\/1970384"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127401003358"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127401002900"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127402004541"},{"key":"rf5","doi-asserted-by":"crossref","first-page":"637","DOI":"10.3934\/dcds.2005.13.637","volume":"13","author":"Ban J. 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