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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.

MSC:
54H20 Topological dynamics (MSC2010)
28D10 One-parameter continuous families of measure-preserving transformations
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[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
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