Patterson--Sullivan distributions in higher rank

TL;DR

This paper extends the concept of Patterson--Sullivan distributions to higher rank symmetric spaces, analyzing eigenfunctions and their quantum limits using advanced calculus and boundary value techniques, contributing to quantum ergodicity understanding.

Contribution

It introduces a new framework for Patterson--Sullivan distributions in higher rank spaces, generalizing previous results from hyperbolic surfaces and linking boundary values to quantum limits.

Findings

Constructed Patterson--Sullivan distributions on Weyl chambers.

Showed these distributions are asymptotic to quantum limits.

Established invariance properties relevant to quantum ergodicity.

Abstract

For a compact locally symmetric space $\XG$ of non-positive curvature, we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators. Using an $h$-\psdiff\ calculus on $\XG$, we define and study lifted quantum limits as weak$^*$-limit points of Wigner distributions. The Helgason boundary values of the eigenfunctions allow us to construct Patterson--Sullivan distributions on the space of Weyl chambers. These distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties, which makes them useful in the context of quantum ergodicity. Our results generalize results for compact hyperbolic surfaces obtained by Anantharaman and Zelditch.