Bounded orbits of certain diagonalizable flows on $SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z})$

TL;DR

This paper proves that the set of points with bounded orbits under specific diagonalizable flows on a certain homogeneous space forms a hyperplane absolute winning set, indicating a strong form of largeness and robustness.

Contribution

It establishes that bounded orbits under these flows constitute a hyperplane absolute winning set, a novel result linking dynamics and geometric measure theory.

Findings

Bounded orbits form a hyperplane absolute winning set

The result applies to certain diagonalizable flows on $SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z})$

The set of such points is large and robust in the space

Abstract

We prove that the set of points that have bounded orbits under certain diagonalizable flows is a hyperplane absolute winning subset of $SL_{n}(\mathbb{R})/SL_{n}(\mathbb{Z})$.