Inverse limits and full families
As first noticed by \textit{R. F. Williams} [Topology 6, 473--487 (1967; Zbl 0159.53702)] and \textit{L. Block} [Trans. Am. Math. Soc. 214, 403--413 (1975; Zbl 0293.57016)] there is a connection between inverse limits and attractors of dynamical systems. Different authors studied since the relationship between the dynamics of the bonding map and the topology of the resulting inverse limit. Many of these results are based on the tent or logistic familly of maps which are both well understood families of interval maps. The present paper gives results for inverse limits on the compact unit interval taking a single bonding map from a full family of one-parameter \(C^1\) unimodal maps \(f_\lambda\). The kneading theory and renormalization operator introduced by Feigenbaum are used in the proofs. It is shown that for \(\lambda\) such that the kneading sequence of \(f_\lambda\) is below that of \(f_{\lambda _\infty}\) where \(\lambda _\infty\) is the Feigenbaum value, the inverse limit with bonding map \(f_\lambda\) is hereditarily decomposable with topological \(\sin(\frac 1x)\)-curves as the dominant subcontinua. Approaching \(\lambda _\infty\) from above similar appearance of Brouwer-Janiszewski-Knaster indecomposable continuum is observed.
scientific article; zbMATH DE number 2092801
scientific article