Bounding entropy for one-parameter diagonal flows on ${\mathrm{SL}}_{\mathtt{d}}(\mathbb{R})/{\mathrm{SL}}_{\mathtt{d}}(\mathbb{Z})$ using linear functionals

Abstract

We give a method to bound the entropy of measures on ${\mathrm{SL}}_{\mathtt{d}}(\mathbb{R})/{\mathrm{SL}}_{\mathtt{d}}(\mathbb{Z})$, which are invariant under a one-parameter diagonal subgroup, in terms of entropy contributions from the regions of the cusp corresponding to different parabolic groups. These bounds depend on an auxiliary linear functional on the Lie algebra of the Cartan group. In follow-up papers, we will show how to optimize this functional to get good bounds on the cusp entropy and prove that in many cases these bounds are sharp.