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

Abstract

We give a method to bound the entropy of measures on SLd(R)/ SLd(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 a 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.