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Boris Solomyak

  1. Hausdorff dimension for fractals invariant under the multiplicative integers.

    Mon, 02/28/2011 - 16:01 — SomNambul
    Authors: Yuval Peres, Boris Solomyak, Richard Kenyon
    Subjects: Dynamical Systems
    Abstract

    We consider subsets of the (symbolic) sequence space that are invariant under
    the action of the semigroup of multiplicative integers. A representative
    example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$
    for all $k$. We compute the Hausdorff and Minkowski dimensions of these sets
    and show that they are typically different. The proof proceeds via a
    variational principle for multiplicative subshifts.

  2. Multifractal structure of Bernoulli convolutions.

    Wed, 11/10/2010 - 16:01 — SomNambul
    Authors: Thomas Jordan, Boris Solomyak, Pablo Shmerkin
    Subjects: Dynamical Systems
    Abstract

    Let $\nu_\lambda^p$ be the distribution of the random series
    $\sum_{n=1}^\infty i_n \lambda^n$, where $i_n$ is a sequence of i.i.d. random
    variables taking the values 0,1 with probabilities $p,1-p$. These measures are
    the well-known (biased) Bernoulli convolutions.

    In this paper we study the multifractal spectrum of $\nu_\lambda^p$ for
    typical $\lambda$. Namely, we investigate the size of the sets

    \[ \Delta_{\lambda,p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log
    \nu_\lambda^p(B(x,r))}{\log r} =\alpha\right\}. \]

  3. Pisot family self-affine tilings, discrete spectrum, and the Meyer property.

    Tue, 02/02/2010 - 16:01 — SomNambul
    Authors: Boris Solomyak, Jeong-Yup Lee
    Subjects: Dynamical Systems
    Abstract

    We consider self-affine tilings in the Euclidean space and the associated
    tiling dynamical systems, namely, the translation action on the orbit closure
    of the given tiling. We investigate the spectral properties of the system.

  4. Pure Point Dynamical and Diffraction Spectra.

    Tue, 10/27/2009 - 16:01 — SomNambul
    Authors: Boris Solomyak, Jeong-Yup Lee, Robert V. Moody
    Subjects: Dynamical Systems
    Abstract

    We show that for multi-colored Delone point sets with finite local complexity
    and uniform cluster frequencies the notions of pure point diffraction and pure
    point dynamical spectrum are equivalent.

  5. Consequences of Pure Point Diffraction Spectra for Multiset Substitution Systems.

    Mon, 10/26/2009 - 14:11 — SomNambul
    Authors: Boris Solomyak, Jeong-Yup Lee, Robert V. Moody
    Subjects: Metric Geometry
    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 $Q$.

  6. Quasisymmetric conjugacy between quadratic dynamics and iterated function systems.

    Mon, 08/17/2009 - 18:30 — SomNambul
    Authors: Kemal Ilgar Eroğlu, Steffen Rohde, Boris Solomyak
    Subjects: Dynamical Systems
    Abstract

    We consider linear iterated function systems (IFS) with a constant
    contraction ratio in the plane for which the ``overlap set'' $\Ok$ is finite,
    and which are ``invertible'' on the attractor $A$, the sense that there is a
    continuous surjection $q: A\to A$ whose inverse branches are the contractions
    of the IFS. The overlap set is the critical set in the sense that $q$ is not a
    local homeomorphism precisely at $\Ok$. We suppose also that there is a
    rational function $p$ with the Julia set $J$ such that $(A,q)$ and $(J,p)$ are
    conjugate.

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