On the failure of Ornstein theory in the finitary category

TL;DR

This paper demonstrates that key classification theorems in ergodic theory do not hold in the finitary setting, showing fundamental limitations of finitary analogues for Bernoulli shifts and related properties.

Contribution

It provides counterexamples and proves the failure of finitary versions of classical theorems, refuting previous conjectures and establishing new limitations in the finitary category.

Findings

Finitary factors of i.i.d. processes may not be i.i.d.

No finitary Sinai's factor theorem for ergodic processes.

Finitary weak Pinsker property does not hold.

Abstract

We show the invalidity of finitary counterparts for three classification theorems: The preservation of being a Bernoulli shift through factors, Sinai's factor theorem, and the weak Pinsker property. We construct a finitary factor of an i.i.d. process which is not finitarily isomorphic to an i.i.d. process, showing that being finitarily Bernoulli is not preserved through finitary factors. This refutes a conjecture of M. Smorodinsky [11], which was first suggested by D. Rudolph [7]. We further show that any ergodic system is isomorphic to a process none of whose finitary factors are i.i.d. processes, and in particular, there is no general finitary Sinai's factor theorem for ergodic processes. An immediate consequence of this result is the invalidity of a finitary weak Pinsker property, answering a question of G. Pete and T. Austin [1].