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High entropy measures on the space of lattices with escape of mass

Taehyeong Kim

TL;DR

The paper investigates entropy along the diagonal flow on the space $X_{m+n}$ of unimodular lattices, focusing on escape of mass near the cusp. By applying the uniform variational principle in the parametric geometry of numbers and constructing high-entropy measures supported close to the cusp, it shows that for $(m,n)\neq (1,1)$ there exist $a$-invariant measures with entropy arbitrarily close to $m+n-1$ whose weak$^*$ limits are the zero measure, and for any $h\in[m+n-1,m+n]$ there are sequences with limits whose mass satisfies $\nu(X_{m+n})=h-(m+n-1)$. This confirms the sharpness of the non-escape bound in KKLM17 for all $(m,n)\neq (1,1)$ (complementing Kad12 for $m=n=1$) and provides a continuum of limiting behaviors by entropy interpolation. The results advance understanding of entropy-escape phenomena in higher-rank homogeneous dynamics and connect template-based variational techniques with cusp dynamics.

Abstract

For any diagonal element $a$ with two eigenvalues, we construct a sequence of $a$-invariant probability measures on the space of unimodular lattices with high entropy but converging to the zero measure. This extends the result of Kadyrov [Ergodic Theory Dynam. Systems, 32(1) (2012)].

High entropy measures on the space of lattices with escape of mass

TL;DR

The paper investigates entropy along the diagonal flow on the space of unimodular lattices, focusing on escape of mass near the cusp. By applying the uniform variational principle in the parametric geometry of numbers and constructing high-entropy measures supported close to the cusp, it shows that for there exist -invariant measures with entropy arbitrarily close to whose weak limits are the zero measure, and for any there are sequences with limits whose mass satisfies . This confirms the sharpness of the non-escape bound in KKLM17 for all (complementing Kad12 for ) and provides a continuum of limiting behaviors by entropy interpolation. The results advance understanding of entropy-escape phenomena in higher-rank homogeneous dynamics and connect template-based variational techniques with cusp dynamics.

Abstract

For any diagonal element with two eigenvalues, we construct a sequence of -invariant probability measures on the space of unimodular lattices with high entropy but converging to the zero measure. This extends the result of Kadyrov [Ergodic Theory Dynam. Systems, 32(1) (2012)].
Paper Structure (4 sections, 6 theorems, 37 equations)

This paper contains 4 sections, 6 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Dimension estimates using variational principle
  3. Construction of high entropy measures
  4. Proofs of Theorem \ref{['Thm_main']} and Corollary \ref{['Cor_main']}

Key Result

Theorem 1.1

KKLM17 For any $h>0$ and any sequence $(\mu_k)_{k\geq1}$ of $a$-invariant probability measures on $X_{m+n}$ with $h_{\mu_k}(a)\geq h$, any weak$^\ast$ limit $\mu$ of the sequence satisfies

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Definition 2.1: Template
  • Definition 2.2: Contraction rate of a template
  • Theorem 2.3: Uniform variational principle
  • Definition 2.4: Standard template
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 5 more