An inverted tower of almost 1-1 extensions
In Isr. J. Math. 45, 1-8 (1983; Zbl 0528.28012) the first author and \textit{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.
scientific article; zbMATH DE number 3905438