TY - JOUR

T1 - A metric minimal PI cascade with 2c minimal ideals

AU - Glasner, Eli

AU - Glasner, Yair

N1 - Funding Information:
Again it is easy to check that, restricted to the fiber of X over z0, ûAûA′ = ûA′ for any choice of sets A, A′ ⊂ Y0, as above. This implies that, for A ∕= A′, the minimal idempotents ûA and ûA′ belong to different minimal left ideals in E(X,T). In fact, if theybelongtothesameminimalleftidealthen,inE(X,T),wehaveûAûA′=ûA,which, in turn, will imply the same equality for the restriction on the fiber over z0, leading to the equality ûA = ûA′ on this fiber, which is a contradiction. It follows that E(X, Tφ) has at least 2c minimal left ideals. By cardinality arguments, mi(X, Tφ) ≤ 2c and our proof is complete. □ Acknowledgements. We thank Ethan Akin for suggesting the approach taken in the proof of Theorem 2.4, and Petra Staynova for pointing out the error in [6] and for producing the figure of the equivalent minimal idempotents u and u′. The research of E. G. was supported by a grant of the Israel Science Foundation (ISF 668/13). The research of Y. G. was supported by a grant from the Israel Science Foundation (ISF 2095/15).
Publisher Copyright:
© Cambridge University Press, 2018.

PY - 2020/5/1

Y1 - 2020/5/1

N2 - We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than (where ) 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 , we construct a metric minimal PI-flow with 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 with minimal left ideals. We go on to construct an example of a minimal PI-flow on a compact manifold and a suitable path-wise connected group of a homeomorphism of , such that the flow is PI and has minimal left ideals. Finally, we use this latter example and a theorem of Dirbák to construct a cascade that is PI (of order three) and has minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication 'less than minimal left ideals implies PI', fails.

AB - We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than (where ) 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 , we construct a metric minimal PI-flow with 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 with minimal left ideals. We go on to construct an example of a minimal PI-flow on a compact manifold and a suitable path-wise connected group of a homeomorphism of , such that the flow is PI and has minimal left ideals. Finally, we use this latter example and a theorem of Dirbák to construct a cascade that is PI (of order three) and has minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication 'less than minimal left ideals implies PI', fails.

KW - PI-flow

KW - enveloping semigroup

KW - minimal left ideals

UR - http://www.scopus.com/inward/record.url?scp=85053733023&partnerID=8YFLogxK

U2 - 10.1017/etds.2018.78

DO - 10.1017/etds.2018.78

M3 - Article

AN - SCOPUS:85053733023

VL - 40

SP - 1268

EP - 1281

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 5

ER -