Shift-maximal sequences in function iteration: Existence, uniqueness and multiplicity
Shift-maximal sequences over two symbols arise in the theory of symbolic dynamics of the iteration of one-dimensional one parameter unimodal transformations \(\lambda\) F(x) for \(\lambda\in (0,1)\). Each sequence \({\mathbb{P}}\) may be associated with none, one, or more than one value of the parameter \(\lambda\). We call these values of \(\lambda\) : \(\lambda_{{\mathbb{P}}}\). We review previous theories of these sequences and prove, under certain conditions on F, the existence and uniqueness of \(\lambda_{{\mathbb{P}}}\). We discuss the theory of multiplicity of the \(\lambda_{{\mathbb{P}}}'s\) and give examples. We also discuss a generalized shift-maximal sequence structure and illustrate quadratic convergence in period doubling, different from Feigenbaum's geometric convergence. Finally, we review some of the experimental observations of shift-maximal sequences.
scientific article; zbMATH DE number 4074011
scientific article