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

摘要

We consider the family of self-similar sets A(lambda), attractors of the iterated function system {C; lambdaz - 1,lambdaz + 1}, depending on a parameter lambda in the open unit disc. First we study the set M of those lambda for which A(lambda) 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 MR is contained in the closure of its interior). Next we turn to the sets A(lambda) themselves and natural measures v(lambda) 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 v(lambda). Next we investigate the Hausdorff dimension and measure of A(lambda), for Lebesgue-a.e. lambda is an element of M, and obtain partial results on the absolute continuity of v(lambda), for a.e. lambda with lambda > 1/root2. [References: 30]