Simultaneous dense and nondense orbits for toral diffeomorphisms

TL;DR

This paper demonstrates that for certain pairs of hyperbolic automorphisms and Anosov diffeomorphisms on the 2-torus, the set of points with mixed dense and nondense orbits is dense and uncountable, revealing intricate orbit structures.

Contribution

It establishes the density and uncountability of points with mixed orbit behaviors for pairs of hyperbolic automorphisms and Anosov diffeomorphisms on the 2-torus, including noncommuting pairs.

Findings

Dense, uncountable set of points with mixed orbit behaviors.

Applicability to noncommuting pairs of maps.

Use of Baire Category theorem and geometric constructions.

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 $C^2$-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.