Möbius orthogonality for generalized Morse-Kakutani flows

The Prouhet-Thue-Morse sequence and its generalizations have occured in many settings. ''Morse-Kakutani flow'' refers to Kakutani's 1967 generalization of the Morse minimal flow (1922). These flows are ${\Bbb Z}_2$ skew products of almost one-to-one extensions of the adding machine ($x\rightarrow x+1$ on the $2-adic$ completion of ${\Bbb Z}$). ''Generalized Morse-Kakutani flow'' is a $K$ skew product of similar construct, with base flow a factor of $x\rightarrow x+1$ on the profinite completion of ${\Bbb Z}$ and $K$ any compact group, abelian or not, of countable density. The principal result is that Sarnak's M\"{o}bius Orthogonality Conjecture holds for a restricted class of generalized Morse-Kakutani flows.