Furstenberg entropy spectra of stationary actions of semisimple Lie groups

TL;DR

This paper characterizes the Furstenberg entropy spectra for ergodic stationary actions of semisimple Lie groups like SL(d,R) and their lattices, using advanced theorems and Poisson bundle constructions.

Contribution

It provides a detailed description of entropy spectra for these actions, extending the Nevo-Zimmer theorem and employing novel Poisson bundle methods.

Findings

Determined Furstenberg entropy spectra for SL(d,R) and lattices.

Derived entropy constraints from a refined Nevo-Zimmer theorem.

Constructed Poisson bundles to realize entropy spectra.

Abstract

We determine Furstenberg entropy spectra of ergodic stationary actions of $SL(d,\mathbb{R})$ and its lattices. The constraints on entropy spectra are derived from a refinement of the Nevo-Zimmer projective factor theorem. The realisation part is achieved by means of building Poisson bundles over stationary random subgroups.