On the 'Mandelbrot set' for pairs of linear maps: Asymptotic self-similarity

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Abstract

We continue the investigation of iterated function systems (IFS) {λz, λz + 1} in the complex plane, depending on a parameter λ in the open unit disc. Let Aλ be the attractor, and let denote the connectedness locus, that is, the set of λ for which Aλ is connected. We show that the set is locally asymptotically self-similar and asymptotically similar to the attractor of the IFS {λz - 1, λz, λz + 1}, for certain 'landmark' points on the boundary of . We also study the parameters for which the attractor A λ is tree-like, and prove that there are uncountable many such λ, answering a question of Bandt.

Original languageEnglish
Pages (from-to)1927-1943
Number of pages17
JournalNonlinearity
Volume18
Issue number5
DOIs
StatePublished - 1 Sep 2005
Externally publishedYes

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