Representation by an automaton of continuous functions of the torus
A function \(f\) from the torus into itself is called base-\(p\) representable if there exists \(Zf\subset (\{0, \dots, p-1\}^\mathbb{N})^2\) such that \((x,y)\in Zf\) if and only if, for \(x\) admitting \(x\) as base-\(p\) expansion, \(y\) is a base-\(p\) expansion of \(f(x)\), and such that the first projection of \(Zf\) on \(\{0, \dots, p-1\}^\mathbb{N}\) is onto. The authors prove that a function is base-\(p\) representable if and only if \[ \exists n\in\mathbb{Z} \quad \exists m\in \{0, \dots, p-1\}, \quad f(x)= \left(nx+ {m\over p-1} \right) \bmod p, \] if and only if the function is representable in base-\(p\) by a transducer. An essential tool is that a continuous function on the torus commuting with the multiplication by \(p\) has the form \(f(x)= (ux+ {m\over p-1}) \bmod 1\) [see also \textit{A. Johnson} and \textit{D. J. Rudolph}, Ergodic Theory Dyn. Syst. 12, No. 4, 743-748 (1992; Zbl 0787.58027)]. Some (minor) remarks: The choice of the letter \(p\) for the base might be misleading \((p\) is not necessarily prime); some misprints occurred (first page, replace ``appellé'' by ``appelé''; and in the bibliography reference [JR] (given above) is not complete, and the year is missing in reference [B]). Finally note that reference [M] appeared also with the same title in IEEE Trans. Comput. 43, 752-755 (1994).
scientific article; zbMATH DE number 912507
scientific article