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We consider the "Mandelbrot set" $M$ for pairs of complex linear maps,introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt andothers. It is defined as the set of parameters $\lambda$ in the unit disk suchthat the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ isconnected. We show that a non-trivial portion of $M$ near the imaginary axis iscontained in the closure of its interior (it is conjectured that all non-realpoints of $M$ are in the closure of the set of interior points of $M$). Next weturn to the attractors $A_\lambda$ themselves and to natural measures$\nu_\lambda$ supported on them. These measures are the complex analogs ofmuch-studied infinite Bernoulli convolutions. Extending the results of Erd\"osand Garsia, we demonstrate how certain classes of complex algebraic integersgive rise to singular and absolutely continuous measures $\nu_\lambda$. Next weinvestigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results onthe absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greaterthan $\sqrt{1/2}$.Comment: 22 pages, 5 figure

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