A metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI

Shmuel Glasner

doi:10.1007/bf02757277

A metric minimal PI cascade with minimal ideals

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.78 ◽

2018 ◽

Vol 40 (5) ◽

pp. 1268-1281

Author(s):

ELI GLASNER ◽

YAIR GLASNER

Keyword(s):

Compact Manifold ◽

Minimal Flow ◽

Connected Group ◽

Skew Products ◽

Enveloping Semigroup

We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than $2^{\mathfrak{c}}$ (where $\mathfrak{c}=2^{\aleph _{0}}$) minimal left ideals is proximal isometric (PI). Then we show the existence of various minimal PI-flows with many minimal left ideals, as follows. For the acting group $G=\text{SL}_{2}(\mathbb{R})^{\mathbb{N}}$, we construct a metric minimal PI $G$-flow with $\mathfrak{c}$ minimal left ideals. We then use this example and results established in Glasner and Weiss. [On the construction of minimal skew-products. Israel J. Math.34 (1979), 321–336] to construct a metric minimal PI cascade $(X,T)$ with $\mathfrak{c}$ minimal left ideals. We go on to construct an example of a minimal PI-flow $(Y,{\mathcal{G}})$ on a compact manifold $Y$ and a suitable path-wise connected group ${\mathcal{G}}$ of a homeomorphism of $Y$, such that the flow $(Y,{\mathcal{G}})$ is PI and has $2^{\mathfrak{c}}$ minimal left ideals. Finally, we use this latter example and a theorem of Dirbák to construct a cascade $(X,T)$ that is PI (of order three) and has $2^{\mathfrak{c}}$ minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication ‘less than $2^{\mathfrak{c}}$ minimal left ideals implies PI’, fails.

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Suspensions of topological transformation groups

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700005411 ◽

1990 ◽

Vol 10 (1) ◽

pp. 101-117

Author(s):

David B. Ellis

Keyword(s):

Topological Group ◽

Transformation Group ◽

Main Tool ◽

Transformation Groups ◽

Minimal Flow ◽

Topological Transformation ◽

Enveloping Semigroup ◽

Dynamical Properties ◽

The Relationship

AbstractLet S be a subgroup of a topological group T, and suppose that S acts on a space X. One can form a T-transformation group (X ×sT, T) called the suspension of the S-transformation group (X, S). In this paper we study the relationship between the dynamical properties of (X, S) and those of its suspension when S is syndetic in T. The main tool used in this study is a notion of the group of a minimal flow (X, T) which is sensitive to the topology on the group T. We are able, using this group and the enveloping semigroup to obtain results on which T-transformation groups can be realized as suspensions of S-transformation groups, and give conditions under which the suspension of an equicontinuous S-flow is an equicontinuous T-flow.

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Relativized Weak Mixing of Uncountable Order

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-042-4 ◽

1980 ◽

Vol 32 (3) ◽

pp. 559-566 ◽

Cited By ~ 2

Author(s):

Douglas McMahon

Keyword(s):

Minimal Flow ◽

Weak Mixing ◽

Enveloping Semigroup ◽

Structure Relation ◽

Limit Ordinal ◽

Dense Set

We show that if Y is a metric minimal flow and θ: Y→Z in an open homomorphism that has a section (i.e., a RIM), and if S(θ)= R(θ),then °YΩ contains a dense set of transitive points, where Ω is the first uncountable ordinalYΩ = П{Y:1 ≦ α < Ω and α not a limit ordinal}, and°YΩ = {y ∈ YΩ:θ(yα)= θ(yβ)for 1 ≦ α,β < Ω and α, β not limit ordinals},S(θ) is the relativized equicontinuous structure relation, andR(θ)= {(y1,y2) ∈ Y X Y:θ(y1) = θ(y2)}.We use this to generalize a result of Glasner that a metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI, [5].I would like to thank Professor T. S. Wu for making helpful suggestions, and thank the referee for his time and effort.

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