Topological conjugacy of linear endomorphisms of the 2-torus

Adler, Roy; Tresser, Charles; Worfolk, Patrick

doi:10.1090/S0002-9947-97-01895-3

Topological conjugacy of linear endomorphisms of the 2-torus
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by Roy Adler, Charles Tresser and Patrick A. Worfolk PDF

Trans. Amer. Math. Soc. 349 (1997), 1633-1652 Request permission

Abstract:

We describe two complete sets of numerical invariants of topological conjugacy for linear endomorphisms of the two-dimensional torus, i.e., continuous maps from the torus to itself which are covered by linear maps of the plane. The trace and determinant are part of both complete sets, and two candidates are proposed for a third (and last) invariant which, in both cases, can be understood from the topological point of view. One of our invariants is in fact the ideal class of the Latimer-MacDuffee-Taussky theory, reformulated in more elementary terms and interpreted as describing some topology. Merely, one has to look at how closed curves on the torus intersect their image under the endomorphism. Part of the intersection information (the intersection number counted with multiplicity) can be captured by a binary quadratic form associated to the map, so that we can use the classical theories initiated by Lagrange and Gauss. To go beyond the intersection number, and shortcut the classification theory for quadratic forms, we use the rotation number of Poincaré.

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