Let \((X(g),a)\) be the shift dynamical system, where the phase space \(X(g)\) of this system is the set of all bisequences over a finite symbol set \(\mathcal S\) with \(\mathrm{card }g>1\). The topology of \(X(g)\) is the product topology induced by the discrete topology of \(\mathcal S\). Let \(\Phi(\mathcal S)\) be the set of all continuous transformations of \(X(\mathcal S)\) into \(X(\mathcal S)\) which commute with \(\sigma\). One means of constructing such is to define an arbitrary mapping of blocks of specified length into single symbols and extending this mapping in a natural manner to infinite sequences. It has been shown by M. L. Curtis, the author and R. C. Lyndon that these mappings, composed with powers of the shift, constitute the entire class \(\Phi(\mathcal S)\). This result permits extensive analysis of the class \(\Phi(\mathcal S)\), the subclass \(E(\mathcal S)\) consisting of those members of \(\Phi(\mathcal S)\) which are onto, and the subgroup \(A(\mathcal S)\) of \(E(\mathcal S)\) consisting of those members which are one-to-one maps. Some of these results are the following.
Any finite group is isomorphic to some subgroup of \(A(\mathcal S)\) [Curtis, the author, Lyndon]. If \(\varphi\in E(\mathcal S)\) then there exists an integer \(K(\varphi)\) such that \(\mathrm{card}\ \varphi^{-1}(x) = K(\varphi)\) if \(x\) is bilaterally transitive (which is the case for almost all \(x)\) [A. M. Gleason and L. R. Welch].
If \(\varphi\in E(\mathcal S)\), and \(y\in\varphi^{-1}(x)\) then \(y\) is periodic if and only if \(x\) is periodic. The analogous result holds for almost periodicity, recurrence and transitivity. If \(\mathrm{card}\ \mathcal S\) is a prime and \(p\ge 2\), then there exists no continuous mapping \(\varphi\) such that \(\varphi^p =\sigma\) [L. R. Welch].
If \(\varphi\in E(\mathcal S)\), then the following statements are equivalent: (1) \(\varphi\) is an exactly \(\mu\)-to-one mapping of \(X(\mathcal S)\) onto \(X(\mathcal S)\); (2) \(\varphi\) is open; (3) \(\varphi\) has a cross-section; (4) for each \(x\in X(\mathcal S)\) any two distinct members of \(\varphi^{-1}(x)\) are distal [O. S. Rothaus].