Ergodic properties of folding maps on spheres

TL;DR

This paper studies how sequences of folding maps on spheres can produce dense trajectories, identifying the minimal number of maps needed for such behavior in various dimensions.

Contribution

It characterizes collections of folding maps that generate dense trajectories on spheres and determines the minimal number of maps required.

Findings

Minimal number of folding maps for dense trajectories is d+1.

Characterization of folding map collections producing dense orbits.

Conditions under which trajectories are dense on spheres.

Abstract

We consider the trajectories of points on $\mathbb{S}^{d - 1}$ under sequences of certain folding maps associated with reflections. The main result characterizes collections of folding maps that produce dense trajectories. The minimal number of maps in such a collection is $d+1$.