Orbit structure and (reversing) symmetriesof toral endomorphisms on rational lattices

Michael Baake; Natascha Neumärker; John A. G. Roberts

doi:10.3934/dcds.2013.33.527

Article Contents

2013, Volume 33, Issue 2: 527-553. Doi: 10.3934/dcds.2013.33.527 open access

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Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

1.
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld
2.
School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052

Received: August 2011

Revised: May 2012

Published: February 2013

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Abstract

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension $2$ and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.

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Mathematics Subject Classification: 37E30, 37E15.

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