Sequence entropy of rank one systems

TL;DR

This paper investigates the behavior of sequence entropy in rank one measure-preserving systems along various types of sequences, revealing conditions for it to be infinite or vanish.

Contribution

It establishes new results on when sequence entropy is infinite or zero for subexponential and polynomial sequences in rank one systems.

Findings

Sequence entropy can be infinite along large classes of sequences.

Sequence entropy necessarily vanishes for subexponential sequences under certain growth conditions.

Flexibility results are obtained for polynomial sequences at critical growth thresholds.

Abstract

We study the sequence entropy of rank one measure-preserving systems along subexponential sequences. We prove that the sequence entropy along a large class of sequences can be infinite using Ornstein's probabilistic constructions. Moreover, we show that sequence entropy necessarily vanishes for subexponential sequences if the growth of tower heights remains below certain growth rates, and obtain a flexibility result for polynomial sequences at this critical threshold.