On non-tameness of the Ellis semigroup
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909829332729856 |
|---|---|
| author | Kellendonk, Johannes |
| author_facet | Kellendonk, Johannes |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On non-tameness of the Ellis semigroup Kellendonk, Johannes Dynamical Systems 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. |
| title | On non-tameness of the Ellis semigroup |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2403.07480 |