Boris Solomyak

Dynamics of self-similar tilings

This paper investigates dynamical systems arising from the action by translations on the orbit closures of self-similar and self-affine tilings of

Sixty Years of Bernoulli Convolutions

{\Bbb R}^d

Pure Point Dynamical and Diffraction Spectra

. The main focus is on spectral properties of such systems which are shown to be uniquely ergodic. We establish criteria for weak mixing and pure discrete spectrum for wide classes of such systems. They are applied to a number of examples which include tilings with polygonal and fractal tile boundaries; systems with pure discrete, continuous and mixed spectrum.

Measure and dimension for some fractal families

The distribution νλ of the random series random series Σ±λn is the infinite convolution product of These measures have been studied since the 1930’s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the fundamental open problem: For which λ∈ is νλ, absolutely continuous?

Nonperiodicity implies unique composition for self-similar translationally finite Tilings

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.

Invariant measures on stationary Bratteli diagrams

We study self-similar sets with overlaps, on the line and in the plane. It is shown that there exist self-similar sets that have non-integral Hausdorff dimension equal to the similarity dimension, but with zero Hausdorff measure. In many cases the Hausdorff dimension is computed for a typical parameter value. We also explore conditions for the validity of Falconers formula for the Hausdorff dimension of self- affine sets, and study the dimension of some fractal graphs.

On the morphology of

Let T be a translationally finite self-similar tiling of Rd. We prove that if T is nonperiodic, then it has the unique composition property. More generally, T has the unique composition property modulo the group of its translation symmetries.

gamma

We study dynamical systems acting on the path space of a stationary (non-simple) Bratteli diagram. For such systems we explicitly describe all ergodic probability measures invariant with respect to the tail equivalence relation (or the Vershik map). These measures are completely described by the incidence matrix of the diagram. Since such diagrams correspond to substitution dynamical systems, this description gives an algorithm for finding invariant probability measures for aperiodic non-minimal substitution systems. Several corollaries of these results are obtained. In particular, we show that the invariant measures are not mixing and give a criterion for a complex number to be an eigenvalue for the Vershik map.

-expansions with deleted digits

We investigate the size of the set of reals which can be represented in base y using only the digits 0,1,3. It is shown that this set has Lebesgue measure zero for y 2/5. Our main goal is to prove that it has Lebesgue measure zero for a certain countable subset of

Invariant measures for parabolic ifs with overlaps and random continued fractions

We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no “overlaps.” We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.