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

Abstract

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 $λ$ in the unit disk such that the attractor $A_λ$ of the IFS $\{λz-1, λz+1\}$ 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 $A_λ$ themselves and to natural measures $ν_λ$ supported on them. These measures are the complex analogs of much-studied infinite Bernoulli convolutions. Extending the results of Erdös 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 $λ$ in the set $M$, for Lebesgue-a.e. $λ$. We also obtain partial results on the absolute continuity of $ν_λ$ for a.e. $λ$ of modulus greater than $\sqrt{1/2}$.