An inverted tower of almost 1-1 extensions.

*(English)*Zbl 0567.54026In 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 |

##### Keywords:

minimal flows; common almost 1-1 extension; no common factor; category of pointed flows; inverted tower of almost 1-1 extensions; common pointed factor
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\textit{S. Glasner} and \textit{D. Maon}, J. Anal. Math. 44, 67--75 (1985; Zbl 0567.54026)

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##### References:

[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 |

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[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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