Vol. 9, No. 4, 2016

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Affine hyperbolic toral automorphisms

Colin Thomson and Donna K. Molinek

Vol. 9 (2016), No. 4, 541–549

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.

topological dynamics, chaos, toral automorphism
Mathematical Subject Classification 2010
Primary: 54H20
Secondary: 37B40
Received: 30 January 2014
Accepted: 17 August 2015
Published: 6 July 2016

Communicated by Michael E. Zieve
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