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Tori Approximation of Families of Diagonally Invariant Measures

Omri Nisan Solan, Yuval Yifrach

TL;DR

This work addresses the problem of understanding diagonal (A) invariant measures on the space $X_n$ of unimodular lattices by demonstrating that portions of noncompact $A$-orbits can be approximated with fixed-proportion compact $A$-orbits. The authors develop an effective, Hecke-operator–driven framework, augmented by a Cassels-inspired number-field construction and strong equidistribution results, to approximate generic ergodic measures by compact ones and to realize prescribed ergodic factors in weak limits. They prove the existence of non-ergodic weak limits with specified ergodic components, establish partial and complete mass-approximation phenomena (including convergence to $c m_{X_n}$ for $c\in(0,1]$), and show partial escape of mass for compact $A$-orbits. These results positively answer questions of Shapira, extend the landscape of invariant measures in homogeneous dynamics, and illuminate the interplay between dynamics on $X_n$, Hecke operators, and arithmetic of special number fields. The techniques provide a blueprint for constructing and controlling limit measures in high-rank settings and may enrich understanding of algebraic and spectral properties of diagonal actions on moduli spaces of lattices.

Abstract

We approximate any portion of any orbit of the full diagonal group $A$ in the space of unimodular lattices in $\RR^n$ using a fixed proportion of a compact $A$-orbit. Using those approximations for the appropriate sequence of orbits, we prove the existence of non-ergodic measures which are also weak limits of compactly supported $A$-invariant measures. In fact, given any countably many $A$-invariant ergodic measures, our methods show that there exists a sequence of compactly supported periodic $A$-invariant measures such that the ergodic decomposition of its weak limit has these measures as factors with positive weight. Using the same methods, we prove that any compactly supported $A$-invariant and ergodic measure is the weak limit of the restriction of different compactly supported periodic measures to a fixed proportion of the time. In addition, for any $c\in (0,1]$ we find a sequence of compactly supported periodic $A$-invariant measures that converge weakly to $cm_{X_n}$ where $m_{X_n}$ denotes the Haar measure on $X_n$. In particular, we prove the existence of partial escape of mass for compact $A$-orbits. These results give affirmative answers to questions posed by Shapira in ~\cite{ShapiraEscape}. Our proofs are based on a modification of Shapira's proof in ~\cite{ShapiraEscape} and on a generalization of a construction of Cassels, as well as on effective equidistribution estimates of Hecke neighbors by Clozel, Oh and Ullmo, and a number theoretic construction of a special number field.

Tori Approximation of Families of Diagonally Invariant Measures

TL;DR

This work addresses the problem of understanding diagonal (A) invariant measures on the space of unimodular lattices by demonstrating that portions of noncompact -orbits can be approximated with fixed-proportion compact -orbits. The authors develop an effective, Hecke-operator–driven framework, augmented by a Cassels-inspired number-field construction and strong equidistribution results, to approximate generic ergodic measures by compact ones and to realize prescribed ergodic factors in weak limits. They prove the existence of non-ergodic weak limits with specified ergodic components, establish partial and complete mass-approximation phenomena (including convergence to for ), and show partial escape of mass for compact -orbits. These results positively answer questions of Shapira, extend the landscape of invariant measures in homogeneous dynamics, and illuminate the interplay between dynamics on , Hecke operators, and arithmetic of special number fields. The techniques provide a blueprint for constructing and controlling limit measures in high-rank settings and may enrich understanding of algebraic and spectral properties of diagonal actions on moduli spaces of lattices.

Abstract

We approximate any portion of any orbit of the full diagonal group in the space of unimodular lattices in using a fixed proportion of a compact -orbit. Using those approximations for the appropriate sequence of orbits, we prove the existence of non-ergodic measures which are also weak limits of compactly supported -invariant measures. In fact, given any countably many -invariant ergodic measures, our methods show that there exists a sequence of compactly supported periodic -invariant measures such that the ergodic decomposition of its weak limit has these measures as factors with positive weight. Using the same methods, we prove that any compactly supported -invariant and ergodic measure is the weak limit of the restriction of different compactly supported periodic measures to a fixed proportion of the time. In addition, for any we find a sequence of compactly supported periodic -invariant measures that converge weakly to where denotes the Haar measure on . In particular, we prove the existence of partial escape of mass for compact -orbits. These results give affirmative answers to questions posed by Shapira in ~\cite{ShapiraEscape}. Our proofs are based on a modification of Shapira's proof in ~\cite{ShapiraEscape} and on a generalization of a construction of Cassels, as well as on effective equidistribution estimates of Hecke neighbors by Clozel, Oh and Ullmo, and a number theoretic construction of a special number field.
Paper Structure (15 sections, 17 theorems, 77 equations, 1 figure)

This paper contains 15 sections, 17 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Measures with Predetermined Ergodic Factors
  3. Partial Escape of Mass and Entire Mass Approximations
  4. Method of Proof
  5. Further Research
  6. Acknowledgments
  7. Notation and Preliminaries
  8. Effective Approximations
  9. Hecke Density
  10. Construction of special number field
  11. Application of Hecke operator to compact orbits
  12. Proof of Lemma \ref{['lem: fastApproximation2']}
  13. Approximation of Invariant Measures
  14. Entire Mass Approximations
  15. Distribution of Hecke neighbors of points close to the cusp

Key Result

Theorem 1.1

Let $(\nu_i)_{i=1}^N \subseteq \mathcal{M}(X_n)^A_e$ be a possibly infinite sequence of $A$-invariant ergodic measures. Then there exists $\mu\in \overline{\mathcal{M}(X_n)^A_c}$ such that the ergodic decomposition of $\mu$ has positive weights on the $\nu_i$'s. In particular for any $\mu\in \mathca

Figures (1)

  • Figure 1: Plot of the boxes in Corollary \ref{['cor: Gluing boxes']}. One can see $S_{v_3}$, and in it $S_{v_1}+w_1, S_{v_2}+w_2$, and in them the regions approximated by $f_1, f_2$.

Theorems & Definitions (53)

  • Theorem 1.1
  • remark 1.2: The weights in the ergodic decomposition
  • Corollary 1.3
  • Theorem 1.4
  • remark 1.5: Volume of the compact orbits
  • Conjecture 1.6
  • Conjecture 1.7
  • remark 1.10
  • remark 1.11: Comparison to benoist2007equidistribution and shapira2021translates
  • Definition 2.1: $O$-notations
  • ...and 43 more