Julia sets in parameter spaces

X. Buff*, C. Henriksen

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

303 Downloads (Pure)

Abstract

Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {fb(z) = λz + bz2 + z3}b∈ℂ contains quasi-conformal copies of the quadratic Julia set J(λz + z2). As a corollary, we show that when the Julia set J(λz + z2) is not locally connected (for example when z → λz + z2 has a Cremer point at 0), the bifurcation locus is not locally connected. To our knowledge, this is the first example of complex analytic parameter space of dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers λ of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, contains a dense Gδ subset of S1.

Original languageEnglish
JournalCommunications in Mathematical Physics
Volume220
Issue number2
Pages (from-to)333-375
ISSN0010-3616
DOIs
Publication statusPublished - 2001

Fingerprint

Dive into the research topics of 'Julia sets in parameter spaces'. Together they form a unique fingerprint.

Cite this