A profinite group invariant for hyperbolic toral automorphisms

Lennard F. Bakker; Pedro Martins Rodrigues

doi:10.3934/dcds.2012.32.1965

Article Contents

2012, Volume 32, Issue 6: 1965-1976. Doi: 10.3934/dcds.2012.32.1965

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A profinite group invariant for hyperbolic toral automorphisms

Lennard F. Bakker^1, and
Pedro Martins Rodrigues^2,

1.
Department of Mathematics, Brigham Young University, Provo, Utah
2.
Department of Mathematics, Instituto Superior Técnico, Univ. Tec. Lisboa, Lisboa

Received: February 2011

Revised: July 2011

Published: June 2012

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Abstract

For a hyperbolic toral automorphism, we construct a profinite completion of an isomorphic copy of the homoclinic group of its right action using isomorphic copies of the periodic data of its left action. The resulting profinite group has a natural module structure over a ring determined by the right action of the hyperbolic toral automorphism. This module is an invariant of conjugacy that provides means in which to characterize when two similar hyperbolic toral automorphisms are conjugate or not. In particular, this shows for two similar hyperbolic toral automorphisms with module isomorphic left action periodic data, that the homoclinic groups of their right actions play the key role in determining whether or not they are conjugate. This gives a complete set of dynamically significant invariants for the topological classification of hyperbolic toral automorphisms.

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Mathematics Subject Classification: Primary: 37C15, 37C25, 27C29; Secondary: 11R99, 20E18.

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