A period formula for torus automorphisms

Pages: 1029 - 1048, Volume 9, Issue 4, July 2003

doi:10.3934/dcds.2003.9.1029       Abstract        Full Text (244.8K)       Related Articles

Peter Seibt - Centre de Physique Théorique, CNRS, Marseille, CNRS Luminy, Case 907, F-13288 Marseille Cedex 09, France (email)

Abstract: We determine the order of integer matrices $A \in SL_2(\mathbb Z)$ on lattices $L_N=\frac{1}{N}\mathbb Z^2/\mathbb Z^2$ of $\mathbb Q^2/\mathbb Z^2$, for $N=P_n \equiv $ the number of n-periodic points ( for the particular matrix-action on the rational 2-torus ). The arguments lean heavily on arithmetical properties of ( integer specializations of ) certain Chebychev polynomials.

Keywords:  Discrete dynamical systems, finite orbit structure, torus automorphisms,periods of integer matrix actions, Chebychev polynomials.
Mathematics Subject Classification:  15A36, 58F20.

Received: January 2001;      Revised: October 2002;      Published: April 2003.