Mixing and equipartition for automorphism invariant processes on regular trees

TL;DR

This work studies entropy equipartition for finite-state processes on a $d$-regular tree that are invariant under parity-preserving automorphisms. By exploiting a boundary/horosphere framework and constructing an amenable action on the boundary, the authors transfer entropy theory from amenable groups to the non-amenable tree setting, obtaining almost everywhere equipartition along horospheres. Under exponential $\psi$-mixing, they further establish a Shannon–McMillan–Breiman type limit along metric spheres of even radius, with Gibbs-measure examples (such as Ising at high temperature) demonstrating natural applicability. Collectively, the results extend classical entropy equipartition to non-amenable graphs via boundary techniques and provide new insights into information flow on trees with strong symmetry.

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.

Theorems & Definitions (16)

• theorem 1.1
• theorem 1.2
• theorem 3.1
• theorem 3.2
• theorem 4.1
• theorem 4.2
• theorem 4.3
• theorem 4.4
• theorem 4.5
• theorem 4.6