Classification of Backward Filtrations and Factor Filtrations: Examples from Cellular Automata

TL;DR

This paper studies backward filtrations generated by cellular automata, proving their standardness and exploring their classification under measure-preserving dynamics, revealing different structural results.

Contribution

It introduces the dynamical classification of factor filtrations and applies it to cellular automata, showing new insights into their measure-theoretic properties.

Findings

Backward filtrations from cellular automata are standard.

Dynamical classification reveals different structural properties.

Invariant sigma-algebras under shift influence filtration classification.

Abstract

We consider backward filtrations generated by processes coming from deterministic and probabilistic cellular automata. We prove that these filtrations are standard in the classical sense of Vershik's theory, but we also study them from another point of view that takes into account the measurepreserving action of the shift map, for which each sigma-algebra in the filtrations is invariant. This initiates what we call the dynamical classification of factor filtrations, and the examples we study show that this classification leads to different results.