Sarnak's conjecture for rank-one subshifts

TL;DR

This paper proves Sarnak's conjecture for specific classes of rank-one subshifts with unbounded cutting parameters, expanding understanding of the conjecture's validity in symbolic dynamics.

Contribution

It verifies Sarnak's conjecture for two new classes of rank-one subshifts, including accc and Katok's map generalizations, using techniques from prior research.

Findings

Sarnak's conjecture holds for accc class of rank-one subshifts.

Sarnak's conjecture verified for Katok's map and its generalizations.

Extends the class of systems known to satisfy Sarnak's conjecture.

Abstract

Using techniques developed in \cite{KLR}, we verify Sarnak's conjecture for two classes of rank-one subshifts with unbounded cutting parameters. The first class of rank-one subshifts we consider are called {\em almost complete congruency classes} ({\em accc}), the definition of which is motivated by the main result of \cite{GS}, which implies that, when a rank-one subshift carries a unique non-atomic invariant probability measure, it is accc if it is measure-theoretically isomorphic to an odometer. The second class we consider consists of Katok's map and its generalizations.