Simultaneously bounded and dense orbits for commuting Cartan actions

Abstract

In this paper we prove that the set of points that have bounded orbits under one regular diagonal flow and dense orbits under the other diagonal flow commuting with the first one has full Hausdorff dimension in $X_3=\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$. To explain its application towards the Uniform Littlewood's Conjecture proposed in \cite{BFK25}, we introduce the concept of ``fiberwise nondivergence'' for the action of a cone inside the full diagonal subgroup. Then our main result implies that there exists a dense subset of $X_3$ in which each point has a fiberwise non-divergent orbit under a cone inside the full diagonal subgroup and an unbounded orbit under every diagonal flow.