Chaos for linear fractional transformations of shifts (Q266315)

The notion of chaotic map (one that is topologically transitive and has a dense set of periodic points) is extended to separable Banach spaces (where the usual terminology from functional analysis is to say the operator is hypercyclic and has a dense set of periodic points), and the property of chaos is characterized for a specific class of bounded linear maps. The existence of hypercyclic operators on a Banach space was shown by \textit{S. Rolewicz} [Stud. Math. 32, 17--22 (1969; Zbl 0174.44203)], who showed that \(\lambda B\), where \(B\) denotes the backward shift operator \(B(x_1,x_2,\dots)=(x_2,x_3,\dots)\) on \(\ell^1\), is hypercyclic if and only if \(|\lambda|>1\). Here the case of weighted \(\ell^p\) spaces is considered, defined by a parameter \(p\in[1,\infty)\) and a strictly positive sequence of weights \(v=(v_n)\) via the condition \(\ell^p_v=\{(x_n)\in\mathbb{C}^{\mathbb{N}}\mid \sum_{n=1}^{\infty}| x_n|^pv_n<\infty\}\). The backward shift operator is bounded if \(\sup_n(v_n/v_{n+1})<\infty\). The main result here uses the Taylor expansion (or the functional calculus) to give a meaningful interpretation of \(\frac{aB+b}{cB+d}\) as an operator on \(\ell^p_v\), and for this collection of operators characterizes the property of chaos in terms of the defining parameters \(a,b,c,d\) and weight sequence \(v\).