On the 'Mandelbrot set' for a pair of linear maps and complex Bernoulli convolutions

We consider the family of self-similar sets A_λ, attractors of the iterated function system {ℂ; λZ -1, λz + 1}, depending on a parameter λ in the open unit disc. First we study the set M of those λ for which A_λ is connected. We show that a non-trivial portion of M near the imaginary axis is the closure of its interior (it is conjectured that M\ℝ is contained in the closure of its interior). Next we turn to the sets A_λ themselves and natural measures ν_λ supported on them. These measures are the complex analogues of much-studied infinite Bernoulli convolutions. Extending the results of Erdos and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures ν_λ. Next we investigate the Hausdorff dimension and measure of A_λ, for Lebesgue-a.e. λ ∈ M, and obtain partial results on the absolute continuity of v_λ, for a.e. λ with |λ| > 1/√2.