The complexity threshold for the emergence of Kakutani inequivalence

TL;DR

This paper establishes that linear complexity is the critical threshold for Kakutani inequivalence in minimal subshifts, demonstrating the existence of systems with low super-linear complexity that admit both types of ergodic measures.

Contribution

It proves that linear complexity is necessary for Kakutani inequivalence and constructs examples of minimal subshifts with low super-linear complexity exhibiting diverse ergodic measures.

Findings

Linear complexity is the threshold for Kakutani inequivalence.

Existence of minimal subshifts with low super-linear complexity admitting different ergodic measures.

No minimal subshift with linear complexity admits Kakutani inequivalence.

Abstract

We show that linear complexity is the threshold for the emergence of Kakutani inequivalence for measurable systems supported on a minimal subshift. In particular, we show that there are minimal subshifts of arbitrarily low super-linear complexity that admit both loosely Bernoulli and non-loosely Bernoulli ergodic measures and that no minimal subshift with linear complexity can admit inequivalent measures.