Times two, three, five orbits on $\mathbb{T}^2$

TL;DR

This paper investigates the density of orbits on the 2-torus under multiplicative actions by 2, 3, and 5, establishing that non-rational points have dense orbits under these transformations.

Contribution

It proves that points not on rational lines have dense orbits under the combined imes 2, 3, and 5 actions on the 2-torus.

Findings

Orbits of non-rational points are dense under the specified actions.

Rational points have non-dense, structured orbits.

The result extends understanding of orbit closures under diagonal torus actions.

Abstract

In this paper, we study orbit closures under diagonal torus actions. We show that if $(x,y)\in\mathbb{T}^2$ is not contained in any rational lines, then its orbit under the $\times 2, \times 3, \times 5$ actions is dense in $\mathbb{T}^2.$