(Non)invertibility of fibonacci-like unimodal maps restricted to their critical omega-limit sets
Henk Bruin · Surrey Research Insight Open Access (The University of Surrey) · 2011
A Fibonacci(-like) unimodal map is defined by special combinatorial properties which guarantee that the critical omega-limit set ω(c) is a minimal Cantor set. In this paper we give conditions to ensure that f |ω(c) is invertible, except at a subset of the backward critical orbit. Furthermore, any finite subtree of the binary tree can appear for some f as the tree connecting all points at which f |ω(c) is noninvertible. This technique gives a new way of finding strange adding machines, i.e., nonrenormalizable maps for which f : ω(c) → ω(c) is conjugate to a (triadic) adding machine. This construction of strange adding machines is compatible with ω(c) being a wild attractor.