On the dynamics of non-reducible cylindrical vortices

TL;DR

This paper investigates the dynamics of cylindrical vortices, a class of skew-maps with minimal base homeomorphisms and affine isometry cocycles, revealing they are never minimal but can be topologically transitive, with new examples constructed.

Contribution

It extends classical results to cylindrical vortices, showing their non-minimality and providing novel examples of topologically transitive vortices.

Findings

No cylindrical vortex is minimal.

Existence of topologically transitive cylindrical vortices.

Extension of classical dynamical results to this new setting.

Abstract

We study skew-maps given by a minimal homeomorphism on the basis and a cocycle of affine isometries on the fibers. We call such a map a cylindrical vortex. We extend to this setting some classical results of Atkinson, Besicovitch, Matsumoto-Shishikura and Schinelman (among other people) about cylindrical cascades. In particular, we show that no cylindrical vortex is minimal, and we construct interesting examples of topologically transitive ones.