Julia sets in parameter spaces

Given a complex number lambda of modulus 1, we show that the bifurcation lo cus of the one parameter family {f(b)(z) = lambdaz + bz(2) + z(3)} b is an element ofC contains quasi-conformal copies of the quadratic Julia set J(la mbdaz + z(2)). As a corollary, we show that when the Julia set J(lambdaz z(2)) is not locally connected (for example when z --> lambdaz + 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 o f dimension 1, with connected but non-locally connected bifurcation locus. We also show that the set of complex numbers lambda of modulus 1, for which at least one of the parameter rays has a non-trivial accumulation set, con tains a dense G(delta) subset of S-1.