Conjugacy problems for hyperbolic automorphisms

Ale, Samson Olatunji (1978) Conjugacy problems for hyperbolic automorphisms. University of Southampton, Doctoral Thesis.

Abstract

The mathematical theory of hyperbolic tonal automorphisms displays many interesting facets. In this thesis we shall investigate the parts of the theory which involve Banach spaces, topology, algebraic number theory, and combinatorial group theory. As maps of the torus, hyperbolic toral automorphisms are structurally and topologically stable and since they have infinitely many periodic points, they provide the simplest examples of structurally stable diffeomorphisms which are not Morse-Smale. Our work shows that for automorphisms f of the 2-torus we may give an explicit neighbourhood of f in the C1-topology, all of whose elements are topologically conjugate to f. Working in the space of homeomorphisms, we also give an explicit size to the C°-neighbourhood of f which occurs in Walters' definition of the topological stability of f. For the torus, the C°and C1 - topologies are metric topologies and the neighbourhoods we find are given in terms of metric distances which depend only on the matrix representation of f.After this discussion of the stability of hyperbolic toral automophisms we turn to the question of deciding when two of them are conjugate. This question is equivalent to the problem of integral similarity between integer matrices. For (2 x 2) matrices we give a detailed investigation of the information contained in the known general theory. In particular we give an explicit method for deciding when two hyperbolic integer matrices are integrally similar. The subtleties of the algebraic results demonstrate the difficulties in obtaining complete topological (dynamical) invariants for conjugacy.

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