, Volume 220, Issue 2, pp 333-375

Julia Sets in Parameter Spaces

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Given a complex number λ of modulus 1, we show that the bifurcation locus of the one parameter family {f b (z)=λz+b z 2+z 3} b contains quasi-conformal copies of the quadratic Julia set Jz+z 2). As a corollary, we show that when the Julia set Jz+z 2) is not locally connected (for example when z↦λz+z 2 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 G δ subset of S 1.

Received: 22 September 2000 / Accepted: 16 January 2001