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Minimal homeomorphisms on low-dimensional tori

Published online by Cambridge University Press:  09 February 2009

N. M. DOS SANTOS  and
R. URZÚA-LUZ
N. M. DOS SANTOS
Affiliation:
Universidade Federal Fluminense, 24020-005 Niteroi, R.J., Brazil (email: [email protected])
R. URZÚA-LUZ
Affiliation:
Universidad Católica de Norte, Casilla 1280, Antofagasta, Chile (email: [email protected])
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Abstract

We study minimal homeomorphisms (all orbits are dense) of the tori Tn, n≤4. The linear part of a homeomorphism φ of Tn is the linear mapping L induced by φ on the first homology group of Tn. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n≤4, then L is quasi-unipontent, that is, all of the eigenvalues of L are roots of unity and conversely if L∈GL(n,ℤ) is quasi-unipotent and 1 is an eigenvalue of L, then there exists a C∞ minimal skew-product diffeomorphism φ of Tn whose linear part is precisely L. We do not know whether these results are true for n≥5. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 5 , October 2009 , pp. 1515 - 1528
Copyright
Copyright © Cambridge University Press 2009

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