Backward \(\varPhi\)-shifts and universality (Q1779347)

A continuous mapping \(T\) from a topological space \(X\) to itself is said to be universal, with a universal element \(x\in X\), provided that the orbit \(\{T^nx:n\geq1\}\) is dense in \(X\). When \(X\) is a real or a complex topological vector space and \(T\) is a bounded linear operator, the term ``universal'' is often replaced by ``hypercyclic''. For a summary of the concepts, a historical account, and statements dealing with universality and hypercyclicity, the interested reader may consult, for instance, the paper by \textit{K.-G. Grosse-Erdmann} [Bull. Am. Math. Soc. 36, 345--381 (1999; Zbl 0933.47003)]. Let \(E\) be a topological space and let \(S\) be a subspace of \(E^{\mathbb{N}}\). A backward \(\varPhi\)-shift on \(S\) is a map \(B_f:S\to S\) defined by \[ B_f((x_n)_{n\geq1}):=(f(x_2),f(x_3),f(x_4),\dots), \] where \(f:E\to E\) is a continuous selfmap. The corresponding \(\varPhi\)-product map on \(S\) is given by \[ \varPi_f((x_n)_{n\geq1}):=(f(x_n))_{n\geq1}. \] In this paper, the author studies the universality of \(B_f\) and \(\varPi_f\) in terms of the dynamical properties of the underlying function \(f\). He gives some applications to the theory of hypercyclic operators. In particular, he extends Rolewicz's theorem on the hypercyclicity of scalar multiples of the classical backward shift.