# zbMATH — the first resource for mathematics

A natural family of factors for minimal flows. (English) Zbl 0916.54024
Nerurkar, M. G. (ed.) et al., Topological dynamics and applications. A volume in honor of Robert Ellis. Proceedings of a conference in honor of the retirement of Robert Ellis, Minneapolis, MN, USA, April 5–6, 1995. Providence, RI: American Mathematical Society. Contemp. Math. 215, 19-42 (1998).
Given a dynamical system (topological or measure theoretic), its family of factors have a rich structure which plays an important role in the ‘structure theorems’ for such systems. Motivated by the work of A. del Junco, M. Lemańczyk, and M. K. Mentzen [Stud. Math. 112, No. 2, 141-164 (1995; Zbl 0814.28007)] in the ergodic theoretic setting, the authors develop notion(s) of a ‘natural family’ of factors for a topological dynamical system (t.d.s). Given a minimal t.d.s. $$(X, T)$$, a self joining of $$(X, T)$$ is any minimal subset of $$(X \times X, T)$$. Analogous to the measure theoretic case, the authors prove the existence of a unique smallest natural family $${\mathcal N}$$ of factors which includes all factors arising from self joinings. The family $${\mathcal N}$$ has the property that given any factor $$Y$$ of $$X$$, there exists a unique natural cover $$\widetilde Y\in{\mathcal N}$$ of $$Y$$ such that the map $$\widetilde Y\to Y$$ is regular (unlike the measure theoretic case this need not be a group extension). The authors characterize the least member of this family as the unique maximal regular factor of $$(X,T)$$. Replacing the notion of self-joining in the above set-up by a more general notion of a ‘B-set’, the authors develop an analogous notion of the corresponding ‘smallest natural family of factors’ and characterize its least element as a Kronecker factor. Analogous notions are developed in the R. Ellis’s algebraic setting (involving $$\tau$$-closed Ellis groups of minimal flows) and as a result a new structure theorem for ‘normal extensions’ of minimal flows is proved. Finally, these results are used to prove another structure theorem for a much larger class of ‘semi-normal flows’.
For the entire collection see [Zbl 0882.00043].

##### MSC:
 54H20 Topological dynamics (MSC2010)
##### Keywords:
minimal flows; self-joining