publications

2024

1. Multivariate mean equicontinuity for finite-to-one topomorphic extensions

   Jonas Breitenbücher, Lino Haupt, and Tobias Jäger

   2024

   Abs arXiv

   In this note, we generalise the concept of topo-isomorphic extensions and define finite topomorphic extensions as topological dynamical systems whose factor map to the maximal equicontinuous factor is measure-theoretically at most m-to-one for some m ∈ ℕ. We further define multivariate versions of mean equicontinuity, complementing the notion of multivariate mean sensitivity introduced by Li, Ye and Yu, and then show that any m-to-one topomorphic extension is mean (m+1)-equicontinuous. This falls in line with the well-known result, due to Downarowicz and Glasner, that strictly ergodic systems are isomorphic extensions if and only if they are mean equicontinuous. While in the multivariate case we can only conjecture that the converse direction also holds, the result provides an indication that multivariate equicontinuity properties are strongly related to finite extension structures. For minimal systems, an Auslander-Yorke type dichotomy between multivariate mean equicontinuity and sensitivity is shown as well.

2023

1. Construction of smooth isomorphic and finite-to-one extensions of irrational rotations which are not almost automorphic

   Lino Haupt, and Tobias Jäger

   2023

   Abs arXiv

   Due to a result by Glasner and Downarowicz, it is known that a minimal system is mean equicontinuous if and only if it is an isomorphic extension of its maximal equicontinuous factor. The majority of known examples of this type are almost automorphic, that is, the factor map to the maximal equicontinuous factor is almost one-to-one. The only cases of isomorphic extensions which are not almost automorphic are again due to Glasner and Downarowicz, who in the same article provide a construction of such systems in a rather general topological setting. Here, we use the Anosov-Katok method in order to provide an alternative route to such examples and to show that these may be realised as smooth skew product diffeomorphisms of the two-torus with an irrational rotation on the base. Moreover - and more importantly - a modification of the construction allows to ensure that lifts of these diffeomorphism to finite covering spaces provide novel examples of finite-to-one topomorphic extensions of irrational rotations. These are still strictly ergodic and share the same dynamical eigenvalues as the original system, but show an additional singular continuous component of the dynamical spectrum.

2022

1. The Hierarchy of Topological Dynamical Systems with Discrete Spectrum (Master’s Thesis)

   Lino Haupt

   Nov 2022

   Revised Form

   Abs PDF

   Equicontinuous dynamical systems are "non-chaotic" in many ways. For example their states continuously depend on initial conditions and have zero entropy. In the case of Abelian acting groupseach point exhibits the same dynamical behavior. This means that any two points can be exchanged by symmetries, i.e. automorphisms of the equicontinuous system. There is a complete classification of Abelian equicontinuous systems as group compactifications. In the non-Abelian case the situation requires more sophisticated tools but also justifies equicontinuous systems to be interpreted as "simple". It is well-known that any (compact) topological dynamical system has an equicontinuous factor maximal among all equicontinuous factors. The invertibility properties as well as other regularity properties of the factor map onto this maximal equicontinuous factor (MEF) can tell us how far a system is away from the simple equicontinuous case. In fact, the literature offers a variety of equivalences between dynamical properties on the one hand and the invertibility properties of the factor map to the MEF on the other hand. Those invertibility properties can be put into a canonical hierarchy starting from homeomorphy. Via those aforementioned equivalences, a parallel hierarchy of dynamical properties starting with equicontinuity emerges. This hierarchy can also be understood as a map of the many different topological models for a measure preserving dynamical system with discrete spectrum. Our aim here is to present a collective overview of the current state of knowledge.