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

Lino Haupt, Tobias Jäger

TL;DR

The paper constructs smooth, mean-equicontinuous isomorphic extensions of an irrational rotation by realizing them as $ C^ fty$-diffeomorphisms of $bT^2$ that are skew products over an irrational base $R_eta$. Using the Anosov–Katok method, the authors produce a limit diffeomorphism with MEF equal to the base rotation and show that, by passing to finite coverings, one obtains $m$-to-$1$ topological extensions that retain the same discrete spectrum as the base while acquiring a singular continuous spectral component. They further ensure total strict ergodicity of all lifts and demonstrate that no new eigenvalues arise, separating the discrete and continuous parts of the spectrum. By carefully controlling cyclic approximations, they realize singular continuous spectrum in the lifted systems, thereby providing explicit smooth, non-almost-automorphic examples of isomorphic extensions with finite-to-one structure and a richer spectral type.

Abstract

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.

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

TL;DR

The paper constructs smooth, mean-equicontinuous isomorphic extensions of an irrational rotation by realizing them as -diffeomorphisms of that are skew products over an irrational base . Using the Anosov–Katok method, the authors produce a limit diffeomorphism with MEF equal to the base rotation and show that, by passing to finite coverings, one obtains -to- topological extensions that retain the same discrete spectrum as the base while acquiring a singular continuous spectral component. They further ensure total strict ergodicity of all lifts and demonstrate that no new eigenvalues arise, separating the discrete and continuous parts of the spectrum. By carefully controlling cyclic approximations, they realize singular continuous spectrum in the lifted systems, thereby providing explicit smooth, non-almost-automorphic examples of isomorphic extensions with finite-to-one structure and a richer spectral type.

Abstract

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.
Paper Structure (13 sections, 10 theorems, 57 equations)

This paper contains 13 sections, 10 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological and measure-preserving dynamics
  4. Spectral theory of dynamical systems
  5. Mean equicontinuity
  6. The Anosov-Katok method
  7. $G_\delta$-characterisation of strict ergodicity
  8. $G_\delta$-characterisation of mean equicontinuity for skew products
  9. Mean equicontinuous skew products on the torus: Proof of Theorem \ref{['t.meanequi_with_full_fibres']}
  10. Total strict ergodicity for lifts and non-existence of additional eigenvalues
  11. Total strict ergodicity of the lifts
  12. Non-existence of additional eigenvalues
  13. Singularity of the continuous spectral component

Key Result

Theorem 1.1

There exist $\mathcal{C}^\infty$-diffeomorphisms $\varphi$ of the two-torus with the following properties.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: auslander1988minimal
  • Remark 2.2
  • Theorem 2.3: Simultaneous Uniform Ergodic Theorem
  • Theorem 2.4: Halmos--von Neumann, VonNeumann1932OperatorenmethodeHalmosVonNeumann1942OperatorMethodsII
  • Theorem 2.5: KatokStepin1967PeriodicApproximations
  • Theorem 2.6: DownarowiczGlasner2015IsomorphicExtensionsAndMeanEquicontinuity
  • Proposition 2.7: DownarowiczGlasner2015IsomorphicExtensionsAndMeanEquicontinuity
  • Proposition 2.8
  • ...and 5 more