Structure theory of (multivariate) mean equicontinuous minimal systems

Abstract: We review recent progress in the structure theory of topological dynamics, which has led to a classification of mean equicontinuous minimal systems both in the univariate and multivariate case. It was first shown by Downarowicz and Glasner in 2015 (building on previous work by Li, Tu and Ye) that a minimal topological dynamical system is mean equicontinuous if and only if it is uniquely ergodic and a topo-isomorphic extension of its maximal equicontinuous factor (MEF), meaning that the topological factor map is at the same time a measure-theoretic isomorphism between the two systems. Subsequently, further subclasses defined by dynamical properties (namely frequent stability and diam mean equicontinuity) where equivalently characterised by their extension structures over the MEF (almost one-to-one and regular, respectively). Other dynamical properties like nullness or tameness, can then be related to the resulting hierarchy (they both fall in the regular extension case). Moreover, a topological invariant named amorphic complexity turned out to be taylor-made for the application to mean equicontinuous systems and allows to distinguish conjugacy classes of a large number of examples, including constant length substitutions and inflation tilings.

More recently, this theory has been successfully extended to the multivariate case: a minimal topological dynamical system is mean m-equicontinuous if and only if it is an m-to-one topomorphic extension of its MEF, and similarly frequent m-stability corresponds to an almost m-to-one extension structure and diam mean m-equicontinuity corresponds to m-regularity. The latter is also implied by m-tameness.

The talk aims to give an overview of these recent advances, which are the result of the efforts of a broader community. Specific results by the speaker were obtained in joint work with Felipe Garcia-Ramos and Xiangdong Ye, Eli Glasner and Christian Oertel, Gabriel Fuhrmann, Maik Gröger and Dominik Kwietniak as well as Jonas Breitenbücher, Lino Haupt and others.

dynamical systemsnumber theory

Audience: researchers in the topic

One World Numeration seminar

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