The boundary of chaos for interval mappings

Clark, Trevor and Trejo, Sofía (2020). The boundary of chaos for interval mappings. Proceedings of the London Mathematical Society, 121(6) pp. 1427–1467.

DOI: https://doi.org/10.1112/plms.12372

Abstract

A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set.

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