Topologically transitive homeomorphisms of quotients of tori

Grant Cairns; Barry Jessup; Marcel Nicolau

doi:10.3934/dcds.1999.5.291

Article Contents

1999, Volume 5, Issue 2: 291-300. Doi: 10.3934/dcds.1999.5.291

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Topologically transitive homeomorphisms of quotients of tori

1.
School of Mathematics, La Trobe University, Melbourne, Australia 3083
2.
Department of Mathematics, University of Ottawa, Ottawa, Canada K1N6N5
3.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra

Revised: December 31, 1997

Published: March 1999

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Abstract

This paper considers the following question: for what finite subgroups $G\subset GL(n, \mathbb Z)$, does there exist an element $A\in GL(n, \mathbb Z)$ inducing a topologically transitive homeomorphism of $T^n$/$G$ We show that for $n = 2$ and 3, the only possibility is $G =\{\pm I\}$. Curiously, in higher dimension the structure is less restrictive. We give a variety of examples in dimension 4. Nevertheless, we show that in dimension $\geq 4$, there are relatively few irreducible examples.

Keywords:

Topologically transitive,
chaos.

Mathematics Subject Classification: 54H20, 57S25.

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