Inverse limits on [0,1] using logistic bonding maps (Q1924654)

Let \(f_\lambda: [0,1]\to [0,1]\) be the logistic map defined by \(f_\lambda(x)= 4\lambda x(1-x)\), for \(0\leq\lambda\leq 1\), and let \(M_\lambda\) denote the inverse limit continuum obtained by using \(f_\lambda\) for each one-step bonding map. The authors investigate the topological structure of \(M_\lambda\) and relate this to known dynamics for the logistic family of maps. For \(0\leq\lambda\leq 1/4\), \(M_\lambda\) is a singleton. For \(1/4<\lambda< \lambda_c\) (\(\approx 0.89249\), the Feigenbaum limit), \(M_\lambda\) is hereditarily decomposable, consisting of a ray limiting to collections of \(\sin(1/x)\)-curves. For \(\lambda= \lambda_c\), \(M_\lambda\) is hereditarily decomposable with only three types of nondegenerate subcontinua: arcs, copies of \(M_\lambda\), and unions of two copies of \(M_\lambda\) intersecting at a common endpoint of a ray. For \(\lambda> \lambda_c\), \(M_\lambda\) always contains a nondegenerate indecomposable continuum. The types and numbers of distinct nondegenerate indecomposable subcontinua present is investigated in terms of the periodic structure of \(f_\lambda\). Throughout, the paper is very well illustrated. While not providing a complete description of \(M_\lambda\) for each parameter value of \(\lambda\), it provides a very useful summary of the close tie between topology and dynamics for this basic family of maps.