Positive Entropy Invariant Measures on the Space of Lattices with Escape of Mass

TL;DR

This paper constructs high-entropy invariant measures on the space of unimodular lattices and demonstrates that their limit measure is zero, revealing insights into the dynamics of lattice spaces with escape of mass.

Contribution

It introduces a sequence of high-entropy invariant measures on unimodular lattices and proves their limit measure is zero, highlighting new phenomena in lattice dynamics.

Findings

Constructed a sequence of invariant measures with high entropy

Proved the limit measure of this sequence is zero

Revealed new behavior of lattice measures with escape of mass

Abstract

On the space of unimodular lattices, we construct a sequence of invariant probability measures under a singular diagonal element with high entropy and show that the limit measure is 0.