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Simultaneous dense and non-dense orbits for toral diffeomorphisms

Published online by Cambridge University Press:  28 January 2016

JIMMY TSENG
JIMMY TSENG*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK email j.tseng@bristol.ac.uk
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Abstract

We show that, for pairs of hyperbolic toral automorphisms on the $2$-torus, the points with dense forward orbits under one map and non-dense forward orbits under the other is a dense, uncountable set. The pair of maps can be non-commuting. We also show the same for pairs of $C^{2}$-Anosov diffeomorphisms on the $2$-torus. (The pairs must satisfy slight constraints.) Our main tools are the Baire category theorem and a geometric construction that allows us to give a geometric characterization of the fractal that is the set of points with forward orbits that miss a certain open set.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 4 , June 2017 , pp. 1308 - 1322
Copyright
© Cambridge University Press, 2016 

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