zbMATH — the first resource for mathematics

An inverted tower of almost 1-1 extensions. (English) Zbl 0567.54026
In Isr. J. Math. 45, 1-8 (1983; Zbl 0528.28012) the first author and B. Weiss have shown that two minimal flows with no common factor need not be disjoint. A question posed by H. Furstenberg was whether two minimal flows exist which have a common almost 1-1 extension (and thus are not disjoint in a very strong sense) and still have no common factor. In what follows we construct a minimal flow (X,T) (X compact metric and \(T: X\to X\) a homeomorphism), with two almost 1-1 factors \(X\to^{\phi_ i}Y_ i\) \((i=1,2)\), such that there are no minimal flow (Z,T) and homomorphisms \(Y_ i\to^{\psi_ i}Z\) with \(\psi_ 1\circ \phi_ 1=\psi_ 2\circ \phi_ 2\). Choosing any point \(x_ 0\in X\) we have, in the category of pointed flows, that \((X,x_ 0)\) is a common almost 1-1 extension of \((Y_ 1,\phi_ 1(x_ 0))\) and \((Y_ 2,\phi_ 2(x_ 0))\) and these latter pointed flows have no non-trivial common pointed factor. This answers a restricted version of Furstenberg’s question. We do not have an answer to the original question. The same flow (X,T) also provides an affirmative answer to a question about the existence of an inverted tower of almost 1-1 extensions, namely, there exists a sequence of almost 1-1 homomorphisms \[ X\to^{\psi_ 1}X_ 1\to^{\psi_ 2}X_ 2\to^{\psi_ 3}... \] such that for every \(x_ 0\in X\) the only common pointed factor of the pointed flows \((X_ n,\psi_ n\circ \psi_{n-1}\circ...\circ \psi_ 1(x_ 0))\) is the trivial flow.

54H20 Topological dynamics (MSC2010)
28D10 One-parameter continuous families of measure-preserving transformations
Full Text: DOI
[1] S. Glasner and B. Weiss,Minimal transformations with no common factor need not be disjoint, Isr. J. Math.45 (1983), 1–8. · Zbl 0528.28012 · doi:10.1007/BF02760665
[2] H. Keynes and J. B. Robertson,Eigenvalue theorems in topological transformation groups, Trans. Am. Math. Soc.139 (1969), 359–369. · Zbl 0176.20602 · doi:10.1090/S0002-9947-1969-0237748-5
[3] T. W. Körner,A counter example concerning recurrent points, to appear.
[4] L. Shapiro,Proximality in minimal transformation groups, Proc. Am. Math. Soc.26 (1970), 521–525. · Zbl 0202.23302 · doi:10.1090/S0002-9939-1970-0266183-2
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.