Approximate transitivity property and Lebesgue spectrum

TL;DR

This paper investigates the approximate transitivity (AT) property in dynamical systems, providing new examples of zero entropy systems without AT, including those with finite spectral multiplicity and systems like Rudin-Shapiro substitution, nil-rotations, and Gaussian systems.

Contribution

It introduces a spectral criterion to identify non-AT systems and presents new examples, expanding understanding of spectral properties related to AT.

Findings

Rudin-Shapiro substitution system is not AT

Certain nil-rotations and Gaussian systems lack AT

Non-AT systems contain a Lebesgue spectral component

Abstract

Exploiting a spectral criterion for a system not to be AT we give some new examples of zero entropy systems without the AT property. Our examples include those with finite spectral multiplicity -- in particular we show that the system arising from the Rudin-Shapiro substitution is not AT. We also show that some nil-rotations on a quotient of the Heisenberg group as well as some (generalized) Gaussian systems are not AT. All known examples of non AT-automorphisms contain a Lebesgue component in the spectrum.