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

We consider the “Mandelbrot set” $M$ for pairs of complex linear maps, introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and others. It is defined as the set of parameters $\lambda$ in the unit disk such that the attractor $A_{\lambda }$ of the IFS $\{\lambda z-1,\lambda z+1\}$ is connected. We show that a non-trivial portion of $M$ near the imaginary axis is contained in the closure of its interior (it is conjectured that all non-real points of $M$ are in the closure of the set of interior points of $M$). Next we turn to the attractors $A_{\lambda }$ themselves and to natural measures ${\nu }_{\lambda }$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erd"os and Garsia, we demonstrate how certain classes of complex algebraic integers give rise to singular and absolutely continuous measures ${\nu }_{\lambda }$. Next we investigate the Hausdorff dimension and measure of $A_{\lambda }$, for $\lambda$ in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on the absolute continuity of ${\nu }_{\lambda }$ for a.e. $\lambda$ of modulus greater than $\sqrt{1\mathrm{/}2}$.