# Generalized Morse-Kaktuani Flows

# Generalized Morse-Kaktuani Flows

The Prouhet-Thue-Morse sequence and its generalizations have occurred in many settings. ``Morse-Kakutani flow'' refers to Kakutani's 1967 generalization of the Morse minimal flow (1922). These flows are $\mathbb{Z}_2$ skew products of almost one-to-one extensions of the adding machine ($x \to x+1$ on the $2$-adic completion of $\mathbb{Z}$). ``Generalized Morse-Kakutani flow'' is a $K$ skew product of similar construct, with base flow *a* factor *of* $x\to x+1$ on the profinite completion $\mathbb{Z}$ and $K$ any compact group of countable density. A review of definitions and some old theorems will be followed by a sketch of a proof that Sarnak's M\"obius Orthogonality Conjecture holds for a restricted class generalized Morse-Kakutani flows. "