Ergodic Description of STIT Tessellations

TL;DR

This paper demonstrates that the renormalized STIT tessellation process exhibits ergodic properties, specifically being a finitary factor of a Bernoulli shift and forming a Bernoulli flow, revealing deep stochastic structure.

Contribution

It establishes the ergodic and Bernoulli properties of the renormalized STIT tessellation process in a rigorous mathematical framework.

Findings

Renormalized process is a finitary factor of a Bernoulli shift.

The continuous-time process forms a Bernoulli flow.

Results apply to all polytopes with nonempty interior.

Abstract

Let (Y_t: t > 0) be the STIT tessellation process. We show that for all polytopes W with nonempty interior and all a>1, the renormalized random sequence (a^n Y_{a^n}: n integer) induced in W, is a finitary factor of a Bernoulli shift. As a corollary we get that the renormalized continuous time process (a^t Y_{a^t}: t real) induced in W is a Bernoulli flow.