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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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##### References:
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