Extensions of Cantor minimal systems and dimension groups (Q2856660)

The authors study relations between the dimension group of a Cantor system \((X,T)\) and the dimension group of its factor \((Y,S)\).NEWLINENEWLINEThey define the cohomology group \(H^n(X|Y)\) of an extension and compare \(H^0(X|Y)\) with \(K_0(X,T)/ p^\ast K_0(Y,S)\), where \(K_0(X,T)\) is the dimension group and \(p^\ast\) is the morphism induced by the factor map \(p\). They obtain characterizations of \(H^n(X|Y)\) for weakly mixing extensions, proximal extensions, and extensions with connected fibers.NEWLINENEWLINEIf \((X,T)\) and \((Y,S)\) are minimal Cantor systems, then \(H^n(X|Y)\) is a torsion group for every \(n\geq 0\) and \(H^0(X|Y)=\text{Torsion}(K_0(X)/p^\ast K_0(Y))\).NEWLINENEWLINEThe papers ends with a appendix, containing a example of application of the theory on Morse minimal systems and a few methods for the construction of minimal extensions.