Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Ron Mor

TL;DR

This work develops a general framework to bound the entropy of $\mathbf{a}$-invariant measures on $SL_{d}(\mathbb{R})/SL_{d}(\mathbb{Z})$ by decomposing cusp contributions along standard parabolic cusps. Central to the method is an auxiliary linear functional $\phi$ on the Cartan algebra, which, together with a careful coding of trajectories into cusp regions, yields finite, tractable upper bounds expressed as a maximum over a finite set of cusp data. The novel aspect is the inclusion of $\phi(\pi_{P}(\alpha^{w}))$ to average entropy across orientations, enabling sharper cusp bounds and simplifying the accounting of complex cusp trajectories in higher rank. The results include explicit bounds on the cusp entropy $h_{\infty}(\mathbf{a})$ and cusp components $h_{\infty,P}(\mathbf{a})$, with proofs leveraging a covering/ Bowen-ball framework and the reductive Borel-Serre compactification to organize boundary behavior. Overall, the approach yields a flexible, computable scheme to bound entropy in high-rank homogeneous dynamics and sets the stage for sharpness results and further optimizations in follow-up 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.

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

TL;DR

This work develops a general framework to bound the entropy of -invariant measures on by decomposing cusp contributions along standard parabolic cusps. Central to the method is an auxiliary linear functional on the Cartan algebra, which, together with a careful coding of trajectories into cusp regions, yields finite, tractable upper bounds expressed as a maximum over a finite set of cusp data. The novel aspect is the inclusion of to average entropy across orientations, enabling sharper cusp bounds and simplifying the accounting of complex cusp trajectories in higher rank. The results include explicit bounds on the cusp entropy and cusp components , with proofs leveraging a covering/ Bowen-ball framework and the reductive Borel-Serre compactification to organize boundary behavior. Overall, the approach yields a flexible, computable scheme to bound entropy in high-rank homogeneous dynamics and sets the stage for sharpness results and further optimizations in follow-up work.

Abstract

We give a method to bound the entropy of measures on 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.
Paper Structure (39 sections, 38 theorems, 356 equations, 2 figures)

This paper contains 39 sections, 38 theorems, 356 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and background
  3. Settings
  4. Cusp neighborhoods
  5. Results
  6. Acknowledgements
  7. Linear subspaces
  8. Metrics on $\operatorname{Gr}(l,\mathtt{d})$ and $\bigwedge^{l}\mathbb{R}^{\mathtt{d}}$
  9. Dynamics on $\operatorname{Gr}(l,\mathtt{d})$
  10. The structure of the cusp in $\operatorname{SL}_{\mathtt{d}}(\mathbb{R})/\operatorname{SL}_{\mathtt{d}}(\mathbb{Z})$
  11. The successive minima of Minkowski
  12. Definitions and basic concepts
  13. Dynamics
  14. The structure of the cusp of $\operatorname{SL}_{\mathtt{d}}(\mathbb{R})/\operatorname{SL}_{\mathtt{d}}(\mathbb{Z})$
  15. $\mathbb{Q}$-parabolic subgroups and flags
  16. ...and 24 more sections

Key Result

Theorem 1.1

For all $\epsilon>0$, there is an open cover $\{U_{P}\}_{P\in\mathcal{P}}$ of $\overline{G/\Gamma}^{\operatorname{RBS}}$ satisfying so that the following holds. For all $\mathtt{a}\in A$ and $\phi\in \operatorname{Lie}(A)^{\ast}$ there is a partition of $G/\Gamma$ with $V_{P,[w]_P}\subseteq U_{P}$ so that for any $\mathtt{a}$-invariant probability measure $\mu$ on $G/\Gamma$, where $D_{\mathtt{

Figures (2)

  • Figure 4.1: An illustration of the construction of $\mathcal{T}_{\delta}$ for $\mathtt{d}=4$.
  • Figure 4.2: An illustration of the construction of $\mathcal{J}_{1}$ for $\mathtt{d}=4$. Final intervals are colored in green, while temporary intervals are colored in red.

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Example
  • Theorem 1.5
  • Theorem 1.6: mor2022bmor2022c
  • Remark 1.7
  • Proposition 2.1
  • Definition 1
  • ...and 90 more