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Colloquium: Renormalization and Decomposability of Inverse Limit Spaces
4:10 p.m. Neill Hall 5W
Jonathon E. Fassett
Abstract: Inverse limit spaces commonly appear as attractors of dynamical systems, thus their topology is of interest. In this talk we investigate the topological structure of 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(1/x)-curves. Approaching the Feigenbaum value from above we see a similar pattern in the appearance of the Brouwer-Janiszewski-Knaster indecomposable continuum.