Zero Entropy Interval Maps And MMLS-MMA Property

TL;DR

This paper proves that interval maps with zero topological entropy generate flows that are minimally mean-attractable and mean-L-stable, confirming Sarnak's conjecture for these systems and providing examples with discrete spectrum.

Contribution

It establishes the MMA and MMLS properties for flows from zero-entropy interval maps, and confirms Sarnak's conjecture in this setting.

Findings

Oscillating sequences are linearly disjoint from these flows

Mobius function is orthogonal to all such flows

Provides an example of a flow with discrete spectrum

Abstract

We prove that the flow generated by any interval map with zero topological entropy is minimally mean-attractable (MMA) and minimally mean-L-stable (MMLS). One of the consequences is that any oscillating sequence is linearly disjoint with all flows generated by interval maps with zero topological entropy. In particular, the M\"obius function is orthogonal to all flows generated by interval maps with zero topological entropy (Sarnak's conjecture for interval maps). Another consequence is a non-trivial example of a flow having the discrete spectrum.