Tomiyama’s K-commutative diagrams of minimal dynamical systems

Abstract

Let [Formula: see text] be the Cantor space and [Formula: see text] be an even-dimensional sphere. By applying a result of the existence of minimal skew products, we show that, associated with any Cantor minimal system [Formula: see text], there is a class [Formula: see text] of minimal skew products on [Formula: see text], such that for any two rigid homeomorphisms [Formula: see text] and [Formula: see text], the notions of approximate [Formula: see text]-conjugacy and [Formula: see text]-strongly approximate conjugacy coincide, which are also equivalent to a [Formula: see text]-version of Tomiyama’s commutative diagram. In fact, this is also the case if [Formula: see text] is replaced by any (infinite) connected finite CW-complex with torsion free [Formula: see text]-group, vanished [Formula: see text]-group and the so-called Lipschitz-minimal-property.