On non-tameness of the Ellis semigroup
Johannes Kellendonk
TL;DR
This paper addresses when the Ellis semigroup $E(X,T)$ of a dynamical system is non-tame by analyzing its kernel and structure group. It leverages the Rees structure of $\ker E$ and the maximal equicontinuous factor to relate the algebraic size of the structure group ${\mathcal H}$ to dynamical features such as the proximal relation. A main result is a partial converse for minimal systems with abelian $T$: if the proximal relation is not transitive and the quotient $X_{max}/{\mathcal L}^{sing}$ is uncountable, then ${\mathcal H}$ has cardinality $2^{\mathfrak c}$, isolating a large algebraic component behind non-tameness. These findings sharpen our understanding of non-tameness by identifying concrete conditions on singular fibres and their differences that force a huge structure group, providing practical criteria for detecting highly non-tame behavior.
Abstract
The Ellis semigroup of a dynamical system $(X,T)$ is tame if every element is the limit of a sequence (as opposed to a net) of homeomorphisms coming from the $T$ action. This topological property is related to the cardinality of the semigroup. Non-tame Ellis semigroups have a cardinality which is that of the power set of the continuum $2^{\mathfrak c}$.The semigroup admits a minimal bilateral ideal and this ideal is a union of isomorphic copies of a group $\mathcal H$, the so-called structure group of $(X,T)$. For almost automorphic systems the cardinality of $\mathcal H$ is at most $\mathfrak c$, that of the continuum. We show a partial converse for minimal $(X,T)$ with abelian $T$, namely that the cardinality of the structure group is $2^{\mathfrak c}$ if the proximal relation is not transitive and the subgroup generated by differences of singular points in the maximal equicontinuous factor is not open.This refines the above statement about non-tame Ellis semigroups, as it locates a particular algebraic component of the latter which has such a large cardinality.