Simultaneous dense and non-dense orbits for toral diffeomorphisms. (28th January 2016)

• Record Type: Journal Article
• Authors: TSENG, JIMMY
• Is Part Of: Ergodic theory and dynamical systems. Volume 37:Number 4(2017)
• Page Start: 1308
• Page End: 1322
• Publication Date: 2016-01-28
• Subjects: Ergodic theory -- Periodicals; Differentiable dynamical systems -- Periodicals; 515.42
• DOI: 10.1017/etds.2015.80
• Languages: English
• ISSNs: 0143-3857

Abstract

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.