19,391 research outputs found
Absolutely Continuous Convolutions of Singular Measures and an Application to the Square Fibonacci Hamiltonian
We prove for the square Fibonacci Hamiltonian that the density of states
measure is absolutely continuous for almost all pairs of small coupling
constants. This is obtained from a new result we establish about the absolute
continuity of convolutions of measures arising in hyperbolic dynamics with
exact-dimensional measures.Comment: 28 pages, to appear in Duke Math.
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
Multifractal structure of Bernoulli convolutions
Let be the distribution of the random series
, where is a sequence of i.i.d. random
variables taking the values 0,1 with probabilities . These measures are
the well-known (biased) Bernoulli convolutions.
In this paper we study the multifractal spectrum of for
typical . Namely, we investigate the size of the sets
Our main results highlight the fact that for almost all, and in some cases
all, in an appropriate range, is
nonempty and, moreover, has positive Hausdorff dimension, for many values of
. This happens even in parameter regions for which is
typically absolutely continuous.Comment: 24 pages, 2 figure
Absolute continuity of complex Bernoulli convolutions
We prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrised families of self-similar sets and measures in the complex plane, extending earlier results.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Solomyak, Boris. Bar Ilan University; Israe
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