Construction of smooth isomorphic and finite-to-one extensions of irrational rotations which are not almost automorphic
Year: 2024
Type: article
Abstract: Due to a result by Glasner and Downarowicz [Isomorphic extensions and applications. Topol. Methods Nonlinear Anal.48(1) (2016), 321–338], 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 to provide an alternative route to such examples and to show that these may be realized 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 diffeomorphisms 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.
DOI: 10.1017/etds.2024.70
Authors: LINO HAUPT, TOBIAS JÄGER
Institution: Friedrich Schiller University Jena
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,450
OpenAlex ID: W4403140159