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

TL;DR

This paper introduces a method to bound the entropy of measures invariant under one-parameter diagonal flows on the space of lattices, using linear functionals on the Lie algebra to relate entropy contributions from different cusp regions.

Contribution

The paper proposes a novel approach to bound entropy via linear functionals, enabling sharper estimates for invariant measures on $SL_{d}( )$/SL_{d}( ) and setting the stage for optimization and sharpness results.

Findings

Provides a method to bound entropy using linear functionals.

Relates entropy contributions to cusp regions and parabolic subgroups.

Lays groundwork for optimizing bounds and proving sharpness in future work.

Abstract

We give a method to bound the entropy of measures on $SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$ 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.