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July  2003, 9(4): 1029-1048. doi: 10.3934/dcds.2003.9.1029

A period formula for torus automorphisms

Peter Seibt 1,

1. 

Centre de Physique Théorique, CNRS, Marseille, CNRS Luminy, Case 907, F-13288 Marseille Cedex 09, France

Received  January 2001 Revised  October 2002 Published  April 2003

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: Chebychev polynomials., finite orbit structure, Discrete dynamical systems, torus automorphisms, periods of integer matrix actions.
Mathematics Subject Classification: 15A36, 58F2.
Citation: Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029
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