Tori Approximation of Families of Diagonally Invariant Measures

TL;DR

This paper develops a method to approximate and analyze the behavior of diagonal group orbits in the space of unimodular lattices, revealing new phenomena like non-ergodic measures and partial escape of mass.

Contribution

It introduces a novel approximation technique for diagonal orbits and constructs non-ergodic measures as weak limits of periodic measures, answering open questions.

Findings

Existence of non-ergodic measures as weak limits of invariant measures

Construction of sequences of measures converging to scaled Haar measure

Demonstration of partial escape of mass for compact orbits

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.