121 research outputs found
Positive-measure self-similar sets without interior
We recall the problem posed by Peres and Solomyak in Problems on self-similar and self-affine sets; an update. Progr. Prob. 46 (2000), 95â106: can one find examples of self-similar sets with positive Lebesgue measure, but with no interior? The method in Properties of measures supported on fat Sierpinski carpets, this issue, leads to families of examples of such sets
Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees
We present examples of rooted tree graphs for which the Laplacian has
singular continuous spectral measures. For some of these examples we further
establish fractional Hausdorff dimensions. The singular continuous components,
in these models, have an interesting multiplicity structure. The results are
obtained via a decomposition of the Laplacian into a direct sum of Jacobi
matrices
Spectral estimates for two-dimensional Schroedinger operators with application to quantum layers
A logarithmic type Lieb-Thirring inequality for two-dimensional Schroedinger
operators is established. The result is applied to prove spectral estimates on
trapped modes in quantum layers
Limit theorems for self-similar tilings
We study deviation of ergodic averages for dynamical systems given by
self-similar tilings on the plane and in higher dimensions. The main object of
our paper is a special family of finitely-additive measures for our systems. An
asymptotic formula is given for ergodic integrals in terms of these
finitely-additive measures, and, as a corollary, limit theorems are obtained
for dynamical systems given by self-similar tilings.Comment: 36 pages; some corrections and improved exposition, especially in
Section 4; references adde
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
Aperiodic order and pure point diffraction
We give a leisurely introduction into mathematical diffraction theory with a
focus on pure point diffraction. In particular, we discuss various
characterisations of pure point diffraction and common models arising from cut
and project schemes. We finish with a list of open problems.Comment: 14 page
Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies
Delone sets of finite local complexity in Euclidean space are investigated.
We show that such a set has patch counting and topological entropy 0 if it has
uniform cluster frequencies and is pure point diffractive. We also note that
the patch counting entropy is 0 whenever the repetitivity function satisfies a
certain growth restriction.Comment: 16 pages; revised and slightly expanded versio
An estimate for the Morse index of a Stokes wave
Stokes waves are steady periodic water waves on the free surface of an
infinitely deep irrotational two dimensional flow under gravity without surface
tension. They can be described in terms of solutions of the Euler-Lagrange
equation of a certain functional. This allows one to define the Morse index of
a Stokes wave. It is well known that if the Morse indices of the elements of a
set of non-singular Stokes waves are bounded, then none of them is close to a
singular one. The paper presents a quantitative variant of this result.Comment: This version contains an additional reference and some minor change
Irreversible quantum graphs
Irreversibility is introduced to quantum graphs by coupling the graphs to a
bath of harmonic oscillators. The interaction which is linear in the harmonic
oscillator amplitudes is localized at the vertices. It is shown that for
sufficiently strong coupling, the spectrum of the system admits a new continuum
mode which exists even if the graph is compact, and a {\it single} harmonic
oscillator is coupled to it. This mechanism is shown to imply that the quantum
dynamics is irreversible. Moreover, it demonstrates the surprising result that
irreversibility can be introduced by a "bath" which consists of a {\it single}
harmonic oscillator
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