Typical path components in tent map inverse limits

2017/12/03 by Boyland, Philip, de Carvalho, André, Hall, Toby
#37B45 #37E05 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.00739

Abstract

In the inverse limit I_s of a tent map f_s restricted to its core, the set GR of points whose path components are bi-infinite and bi-dense has full measure with respect to the measure induced on I_s by the unique absolutely continuous invariant measure of f_s. With respect to topology, there is a dichotomy. When the parameter s is such that the critical orbit of f_s is not dense, GR contains a dense G_δ set. In contrast, when the critical orbit of f_s is dense, the complement of GR contains a dense G_δ set.

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