Inverse limits and full families

In this paper we investigate inverse limits on [0,1] using a single bonding map chosen from a Full family (one-parameter family of C1 unimodal maps). Our investigation makes use of the renormalization operator utilized by Feigenbaum to explain the universal way in which Full families transition from simple to complicated dynamics. Among other results, we show that up through the Feigenbaum value the inverse limit is hereditarily decomposable with a fascinating pattern in the appearance of topological sin(1x)-curves. Approaching the Feigenbaum value from above we see a similar pattern in the appearance of the Brouwer–Janiszewski–Knaster indecomposable continuum.