[Paper Review] Simultaneous dense and nondense orbits for toral automorphisms

Jimmy Tseng|arXiv (Cornell University)|Jun 8, 2014

Mathematical Dynamics and Fractals3 references3 citations

TL;DR

This paper demonstrates that for pairs of hyperbolic toral automorphisms on the 2-torus, the set of points with dense forward orbits under one map and nondense forward orbits under the other is both dense and uncountable, even when the maps do not commute. The proof relies on a geometric construction and the Baire Category theorem to establish the prevalence of such mixed orbit behavior.

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. Our main tools are a geometric construction and the Baire Category theorem.

Motivation & Objective

• To investigate the coexistence of dense and nondense forward orbits under two distinct hyperbolic toral automorphisms on the 2-torus.
• To determine whether such mixed orbit behavior is prevalent in the topological sense, despite the complexity of dynamical systems on the torus.
• To extend understanding of orbit structure in noncommuting dynamical systems by identifying the prevalence of points with contrasting orbit behaviors.
• To establish the existence of a large, uncountable set of points exhibiting divergent dynamical behavior under two different automorphisms.

Proposed method

• Utilizing a geometric construction to analyze the distribution of orbits under two toral automorphisms simultaneously.
• Applying the Baire Category theorem to prove that the set of points with dense orbits under one map and nondense under the other is dense in the 2-torus.
• Focusing on hyperbolic toral automorphisms, which are linear, Anosov maps with real eigenvalues not equal to ±1.
• Analyzing forward orbits under each map separately and comparing their density properties in the topological space.
• Using the topological structure of the 2-torus and properties of residual sets to establish uncountability of the mixed orbit set.

Experimental results

Research questions

• RQ1 Can points exist that have dense forward orbits under one hyperbolic toral automorphism but nondense forward orbits under another?
• RQ2 Is the set of such points topologically large—specifically, dense and uncountable—on the 2-torus?
• RQ3 Does this phenomenon persist when the two automorphisms do not commute?
• RQ4 What topological tools can be used to prove the prevalence of mixed orbit behavior in dynamical systems on the torus?

Key findings

• The set of points with dense forward orbits under one automorphism and nondense forward orbits under another is dense in the 2-torus.
• This set is uncountable, indicating a rich and complex structure of mixed dynamical behavior.
• The result holds even when the two toral automorphisms do not commute, demonstrating robustness of the phenomenon.
• The Baire Category theorem is instrumental in proving the topological largeness of the set of mixed-orbit points.