The sums of powers theorem for commuting block maps

Abstract:

A block map is a map $f:{\{ 0, 1\} ^n} \to \{ 0, 1\}$ for some $n \geqslant 1$. A block map $f$ induces an endomorphism ${f_\infty }$ of the full $2$-shift $(X, \sigma )$. Composition of block maps is defined in such a way that ${(f \circ g)_\infty } = {f_\infty } \circ {g_\infty }$. In this paper some recent results concerning the set $\{ g|g \circ f = f \circ g\}$ for certain types of block maps $f$ are extended.