Geometry and combinatorics of Julia sets of real quadratic maps (Q1073438)

For real \(\lambda\) a correspondence is made between the Julia set \(B_{\lambda}\) for \(z\to (z-\lambda)^ 2\), in the hyperbolic case, and the set of \(\lambda\)-chains \(\{\lambda \pm \sqrt{(\lambda \pm \sqrt{(\lambda \pm...}}\},\) with the aid of Cremer's theorem. It is shown how a number of features of \(B_{\lambda}\) can be understood in terms of \(\lambda\)-chains. The structure of \(B_{\lambda}\) is determined by certain equivalence classes of \(\lambda\)-chains, fixed by orders of visitation of certain real cycles; and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of \(\lambda\)- chains. The functional equations obeyed by attractive cycles are investigated, and their relation to \(\lambda\)-chains is given. The first cascade of period-doubling bifurcations is described from the point of view of the associated Julia sets and \(\lambda\)-chains. Certain ''Julian sets'' associated with the Feigenbaum function and some theorems of Lanford are discussed.