On Moebius orthogonality for subshifts of finite type with positive topological entropy

TL;DR

This paper demonstrates that Moebius orthogonality fails for subshifts of finite type with positive topological entropy, implying that certain surface diffeomorphisms also do not exhibit orthogonality with the Moebius function.

Contribution

It establishes the non-orthogonality of the Moebius function to a broad class of dynamical systems with positive entropy, extending understanding of their correlation properties.

Findings

Moebius orthogonality does not hold for subshifts of finite type with positive entropy

All $C^{1+eta}$ surface diffeomorphisms with positive entropy correlate with the Moebius function

The result challenges assumptions about randomness and orthogonality in complex dynamical systems.

Abstract

In this note we prove that Moebius orthogonality does not hold for subshifts of finite type with positive topological entropy. This, in particular, shows that all $C^{1+\alpha}$ surface diffeomorphisms with positive entropy correlate with the Moebius function.