TY - JOUR

T1 - On the 'Mandelbrot set' for a pair of linear maps and complex Bernoulli convolutions

AU - Solomyak, Boris

AU - Xu, Hui

PY - 2003/9

Y1 - 2003/9

N2 - We consider the family of self-similar sets Aλ, 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 Aλ 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 Aλ themselves and natural measures νλ 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 νλ. Next we investigate the Hausdorff dimension and measure of Aλ, for Lebesgue-a.e. λ ∈ M, and obtain partial results on the absolute continuity of vλ, for a.e. λ with |λ| > 1/√2.

AB - We consider the family of self-similar sets Aλ, 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 Aλ 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 Aλ themselves and natural measures νλ 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 νλ. Next we investigate the Hausdorff dimension and measure of Aλ, for Lebesgue-a.e. λ ∈ M, and obtain partial results on the absolute continuity of vλ, for a.e. λ with |λ| > 1/√2.

UR - http://www.scopus.com/inward/record.url?scp=0142168378&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/16/5/311

DO - 10.1088/0951-7715/16/5/311

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SN - 0951-7715

VL - 16

SP - 1733

EP - 1749

JO - Nonlinearity

JF - Nonlinearity

IS - 5

ER -