Affine hyperbolic toral automorphisms

Thomson, Colin; Molinek, Donna K.

doi:10.2140/involve.2016.9.541

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Affine hyperbolic toral automorphisms

Colin Thomson and Donna K. Molinek

Vol. 9 (2016), No. 4, 541–549

DOI: 10.2140/involve.2016.9.541

Abstract

A hyperbolic transformation of the torus is an example of a function that is Devaney chaotic; that is, it is topologically transitive and has dense periodic points. An irrational rotation of the torus, on the other hand, is not chaotic because it has no periodic points. We show that a hyperbolic transformation of the torus followed by a translation (an affine hyperbolic toral automorphism) has dense periodic points and maintains transitivity. As a consequence, affine toral automorphisms are chaotic, even when the translation is an irrational rotation.

Keywords

topological dynamics, chaos, toral automorphism

Mathematical Subject Classification 2010

Primary: 54H20

Secondary: 37B40

Milestones

Received: 30 January 2014

Accepted: 17 August 2015

Published: 6 July 2016

Communicated by Michael E. Zieve

Authors

Colin Thomson
University of North Carolina at Chapel Hill Phillips Hall CB#3250 Chapel Hill, NC 27599 United States

Donna K. Molinek
Davidson College Box 6999 Davidson, NC 28035 United States

Vol. 9, No. 4, 2016