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Let $(X,G)$ be a topological dynamical system, given by the action of a countable discrete infinite group on a compact metric space $X$. We prove that if $(X,G)$ is minimal, then it is either diam-mean $m$-equicontinuous or diam-mean $m$-sensitive. Similarly, $(X,G)$ is either frequently $m$-stable or strongly $m$-spreading. Further, when $G$ is abelian (or, more generally, virtually nilpotent), then the following statements are equivalent: (i) $(X,G)$ is a regular $m$-to-one extension of its maximal equicontinuous factor; (ii) $(X,G)$ is diam-mean $(m+1)$-equicontinuous, and not diam-mean $m$-equicontinuous; (iii) $(X,G)$ is not diam-mean $(m+1)$-sensitive, but diam-mean $m$-sensitive; (iv) $(X,G)$ has an essential weakly mean sensitive $m$-tuple but no essential weakly mean sensitive $(m+1)$-tuple. This provides a “local” characterisation of $m$-regularity and mean $m$-sensitivity via weakly mean sensitive tuples. The same result holds when $G$ is amenable and $(X,G)$ satisfies the local Bronstein condition.

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 $m \in \mathbb{N}$. 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.

In this paper, we introduce and investigate multivariate versions of frequent stability and diam-mean equicontinuity — called “frequent $m$-stability” and “diam-mean $m$-equicontinuity”, where $m > 1$ is a natural number. Here, “multivariate” refers to the dynamical behaviour of tuples of starting points. Systems for which the factor map to the maximal equicontinuous factor (mef) is finite-to-one for a residual set — “almost finite-to-one extensions” — or a set of full measure — “almost-surely finite-to-one extensions” — are respectively characterised by these dynamical rigidity properties.

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.

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 groups each 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.