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1.
Periodica Mathematica Hungarica - 相似文献
2.
Yuval Peres Michal Rams Ká roly Simon Boris Solomyak 《Proceedings of the American Mathematical Society》2001,129(9):2689-2699
A compact set is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the ``open set condition' (OSC), then has positive -dimensional Hausdorff measure, where is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the -dimensional Hausdorff measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting.
3.
Abstract. There is a growing body of results in the theory of discrete point sets and tiling systems giving conditions under which
such systems are pure point diffractive. Here we look at the opposite direction: what can we infer about a discrete point
set or tiling, defined through a primitive substitution system, given that it is pure point diffractive? Our basic objects
are Delone multisets and tilings, which are self-replicating under a primitive substitution system of affine mappings with
a common expansive map Q . Our first result gives a partial answer to a question of Lagarias and Wang: we characterize repetitive substitution Delone
multisets that can be represented by substitution tilings using a concept of ``legal cluster.' This allows us to move freely
between both types of objects. Our main result is that for lattice substitution multiset systems (in arbitrary dimensions),
being a regular model set is not only sufficient for having pure point spectrum—a known fact—but is also necessary.
This completes a circle of equivalences relating pure point dynamical and diffraction spectra, modular coincidence, and model
sets for lattice substitution systems begun by the first two authors of this paper. 相似文献
4.
We prove that a primitive substitution Delone set, which is pure point diffractive, is a Meyer set. This answers a question
of J.C. Lagarias. We also show that for primitive substitution Delone sets, being a Meyer set is equivalent to having a relatively
dense set of Bragg peaks. The proof is based on tiling dynamical systems and the connection between the diffraction and dynamical
spectra.
The first author acknowledges support from the NSERC post-doctoral fellowship and thanks the University of Washington and
the University of Victoria for being the host universities of the fellowship. The second author is grateful to the Weizmann
Institute of Science where he was a Rosi and Max Varon Visiting Professor when this work was completed. He was also supported
in part by NSF Grant DMS 0355187. 相似文献
5.
The paper gives first quantitative estimates on the modulus of continuity of the spectral measure for weak mixing suspension flows over substitution automorphisms, which yield information about the “fractal” structure of these measures. The main results are, first, a Hölder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hölder estimate for self-similar suspension flows; and, third, a Hölder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hölder estimates for the spectral measures of translation flows along stable foliations of pseudo-Anosov automorphisms. A key technical tool in the proof of the second result is an “arithmetic-Diophantine” proposition, which has other applications. In Appendix A this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions. 相似文献
6.
For the eigenvalue problem—λΔu = q(x)u in IRd, with the weight function q changing sign, conditions are discussed for existence of eigenvalues with positive decaying eigenfunctions. 相似文献
7.
B. Solomyak 《Journal of Mathematical Sciences》2007,140(3):452-460
It is proved that every pseudo-self-affine tiling in ?d is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronoï tessellations is a corollary. Previously, these results were obtained in the planar case, jointly with Priebe Frank. The new approach is based on the theory of graph-directed iterated function systems and substitution Delone sets developed by Lagarias and Wang. Bibliography: 18 titles. 相似文献
8.
T. B. Solomyak 《Journal of Mathematical Sciences》1994,72(6):3459-3466
We consider a planar domain, namely a curvilinear quadrilateral. We study a variational inequality of special form on the set of functions that are monotonically increasing on part of the boundary. This problem corresponds to a one-sided problem for an elliptic equation. A boundary condition of first kind is prescribed on part of the boundary, while on the other part of the boundary the tangential derivative is nonnegative and the product of the tangential and oblique derivatives is zero. We establish that the first derivatives of the solution satisfy a Hölder condition. Bibliography: 5 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 173–186. 相似文献
9.
B. M. Solomyak 《Journal of Mathematical Sciences》1992,61(2):2002-2018
On the basis of a functional model one considers the scattering for operators that are close to unitary. In particular cases the presented scheme contains a series of results of L. de Branges and L. Shulman, S. N. Naboko, L. A. Sakhnovich, and H. Neidhardt. The fundamental new result is: the existence of complete local wave operators for a unitary operator and its nuclear perturbation, where the spectrum of the latter does not fill out the unit circle. In this same situation an invariance principle is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 92–119, 1989. 相似文献
10.