Some recent progress on tameness in minimal systems

Gabriel Fuhrmann (Imperial College London)

05-Jun-2020, 08:15-09:45 (4 years ago)

Abstract: Tameness is a notion which--very roughly speaking--refers to the absence of topological complexity of a dynamical system. The last decades saw an increased interest in tame systems revealing their connections to other areas of mathematics like Banach spaces, substitutions and tilings or even model theory and logic. In this talk, we will assume a dynamical systems perspective.

Huang showed that, given a minimal system, tameness implies almost automorphy [1]. That is, after discarding a meagre set of points, the factor map of a tame minimal system to its maximal equicontinuous factor is one-to-one. This structural theorem got recently extended to actions of general groups by Glasner [2].

In a collaboration with Glasner, Jäger and Oertel, we could further improve this result by showing that tame minimal systems are actually regularly almost automorphic [3]. In this talk, we will show a closely related statement which, however, is way easier to prove: every symbolic almost automorphic extension of an irrational rotation whose non-invertible fibres form a Cantor set is non-tame. We will further discuss some related results from a collaboration with Kwietniak [4]. Finally, if time allows, we will come to discuss tameness in substitutive subshifts and more general classes of Toeplitz flows [5].

All (non-standard) notions will be introduced in the talk. In other words: we prioritise accessibility over the number of results to be discussed.

[1] W. Huang, Tame systems and scrambled pairs under an abelian group action, Ergodic Theory Dynam. Systems 26 (2006), 1549-1567.

[2] E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math. 211 (2018), 213-244.

[3] G. Fuhrmann, E. Glasner, T. Jäger, C. Oertel, Irregular model sets and tame dynamics, arXiv:1811.06283, (2018), 1-22.

[4] G. Fuhrmann, D. Kwietniak, On tameness of almost automorphic dynamical systems for general groups, Bull. Lon. Math. Soc. 52 (2020), 24-42.

[5] G. Fuhrmann, J. Kellendonk, R. Yassawi, work in progress.

dynamical systems

Audience: researchers in the topic


Dynamical systems seminar at the Jagiellonian University

Organizers: Dominik Kwietniak, Roman Srzednicki, Klaudiusz Wójcik
Curator: Marcin Kulczycki*
*contact for this listing

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