Simultaneous dense and nondense orbits for toral diffeomorphisms

2014/06/08 by Tseng, Jimmy · 1 citation
#37C05 #37D05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1406.1970

Abstract

We show that, for pairs of hyperbolic toral automorphisms on the 2-torus, the points with dense forward orbits under one map and nondense forward orbits under the other is a dense, uncountable set. The pair of maps can be noncommuting. We also show the same for pairs of C²-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.

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