Mixing and equipartition for automorphism invariant processes on regular trees

TL;DR

This paper investigates the distribution of information in automorphism-invariant processes on regular trees, establishing equipartition results and a Shannon-McMillan-Breiman theorem under mixing conditions.

Contribution

It introduces new equipartition and entropy theorems for invariant processes on regular trees, extending classical results to a tree automorphism setting.

Findings

Almost everywhere equipartition along spheres and horospheres

Shannon-McMillan-Breiman theorem along metric spheres under mixing

Results apply to ergodic automorphism-invariant processes

Abstract

The paper is devoted to equipartition of measured information for finite state processes over regular trees whose laws are invariant under all parity preserving tree automorphisms. We show almost everywhere equipartition for ergodic processes along spheres and balls in every horosphere. Moreover, under a quantitive mixing condition we obtain a Shannon-McMillan-Breiman theorem along metric spheres of even radius.