2,126 research outputs found
On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions
We consider the "Mandelbrot set" 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 in the unit disk such
that the attractor of the IFS is
connected. We show that a non-trivial portion of near the imaginary axis is
contained in the closure of its interior (it is conjectured that all non-real
points of are in the closure of the set of interior points of ). Next we
turn to the attractors themselves and to natural measures
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 . Next we
investigate the Hausdorff dimension and measure of , for
in the set , for Lebesgue-a.e. . We also obtain partial results on
the absolute continuity of for a.e. of modulus greater
than .Comment: 22 pages, 5 figure
Convolutions of Cantor measures without resonance
Denote by the distribution of the random sum , where and all the choices are
independent. For , the measure is supported on , the
central Cantor set obtained by starting with the closed united interval,
removing an open central interval of length , and iterating this
process inductively on each of the remaining intervals.
We investigate the convolutions , where
is a rescaling map. We prove that if the ratio is irrational and , then where denotes any of
correlation, Hausdorff or packing dimension of a measure.
We also show that, perhaps surprisingly, for uncountably many values of
the convolution is a
singular measure, although and is irrational
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