Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-03T19:15:59.345Z Has data issue: false hasContentIssue false
  • English
  • Français

The linearisations of cyclic permutation have rational zeta functions

Published online by Cambridge University Press:  17 April 2009

Bau-Sen Du
Bau-Sen Du
Affiliation:
Institute of Mathematics, Academia Sinica, Taipei, Taiwan 11529, Republic of China, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let n ≥ 2 be an integer. Let P be the set of all integers in [1,n + 1] and let σ be a cyclic permutation on P. Assume that f is the linearisation of σ on P. Then we show that f has rational Artin-Mazur zeta function which is closely related to the characteristic polynomial of some n × n matrix with entries either zero or one. Some examples of non-conjugate maps with the same Artin-Mazur zeta function are also given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 62 , Issue 2 , October 2000 , pp. 287 - 295
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Artin, M. and Mazur, B., ‘On periodic points’, Ann. Math. 81 (1965), 82–99.Google Scholar
[2]Du, B.-S., ‘A simple method which generates infinitely many congruence identities’, Fibonacci Quart. 27 (1989), 116–124.Google Scholar
[3]Du, B.-S., ‘Congruence identities arising from dynamical systems’ Appl. Math. Lett. 12 (1999), 115–119.Google Scholar
[4]Herstein, I.N., Topics in algebra, first edition (Blaisdell Publ. Co., New York, 1964).Google Scholar
[5]Milnor, J. and Thurston, W., On iterated maps of the interval, Lecture Notes in Mathematics 1342 (Springer-Verlag, Berlin, Heidelberg, New York, 1988), 465–563.Google Scholar
[6]Nusse, H.E., ‘Chaotic maps with rational zeta function’, Trans. Amer. Math. Soc. 304 (1987), 705–719.CrossRefGoogle Scholar