Bounded orbits of diagonalizable flows on $\rm{SL}_3(\mathbb{R})/\rm{SL}_3(\mathbb{Z})$

TL;DR

This paper proves that for countably many diagonalizable flows on the space of lattices in three dimensions, the set of lattices with all bounded orbits under these flows has full Hausdorff dimension.

Contribution

It establishes that the set of lattices with bounded orbits under countably many diagonalizable flows has full Hausdorff dimension, extending previous results to multiple flows.

Findings

The set of lattices with bounded orbits under all flows has full Hausdorff dimension.

The result applies to countably many diagonalizable subgroups in SL(3,R).

Bounded orbits are shown to be prevalent in a measure-theoretic sense.

Abstract

We prove that for any countably many one-parameter diagonalizable subgroups $F_n$ of $\rm{SL}_3(\mathbb{R})$, the set of $\Lambda\in\rm{SL}_3(\mathbb{R})/\rm{SL}_3(\mathbb{Z})$ such that all the orbits $F_n\Lambda$ are bounded has full Hausdorff dimension.