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Boris Solomyak

Publications and source records attributed to Boris Solomyak.

At least 37 records · Page 2Linked to original sources
arXiv2019

On substitution tilings and Delone sets without finite local complexity

We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder sets. We then obtain several results on their ergodic-theoretic properties, notably absence of strong mixing and conditions for existence of eigenvalues, which have number-theoretic consequences. In particular, if the set of eigenvalues of the expansion matrix is totally non-Pisot, then the tiling dynamical system is weakly mixing. Further, we define the notion of rigidity for substitution tilings and demonstrate that the result of [Lee-Solomyak (2012)] on the equivalence of four properties: relatively dense discrete spectrum, being not weakly mixing, the Pisot family, and the Meyer set property, extends to the non-FLC case, if we assume rigidity instead.

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arXiv2018

Delone sets and dynamical systems

In these expository notes we focus on selected topics around the themes: Delone sets as models for quasicrystals, inflation symmetries and expansion constants, substitution Delone sets and tilings, and associated dynamical systems.

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arXiv2017

On the dimension of Furstenberg measure for $SL_2(R)$ random matrix products

Let $μ$ be a measure on $SL_{2}(\mathbb{R})$ generating a non-compact and totally irreducible subgroup, let $χ>0$ denote its Lyapunov exponent, and let $ν$ be the associated stationary (Furstenberg) measure for the action on the projective line. We prove that if $μ$ is supported on finitely many matrices with algebraic entries, then \[ \dimν=\min\{1,\frac{h_{\textrm{RW}}(μ)}{2χ}\} \] where $h_{\textrm{RW}}(μ)$ is the random walk entropy of $μ$, and $\dim$ denotes pointwise dimension. In particular, for every $δ>0$, there is a neighborhood $U$ of the identity in $SL_{2}(\mathbb{R})$ such that if a measure $μ\in\mathcal{P}(U)$ is supported on algebraic matrices with all atoms of size at least $δ$, and generates a group which is non-compact and totally irreducible, then its stationary measure $ν$ satisfies $\dimν=1$.

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arXiv2017

On ergodic averages for parabolic product flows

We consider a direct product of a suspension flow over a substitution dynamical system and an arbitrary ergodic flow and give quantitative estimates for the speed of convergence for ergodic integrals of such systems. Our argument relies on new uniform estimates of the spectral measure for suspension flows over substitution dynamical systems. The paper answers a question by Jon Chaika.

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arXiv2017

Singular substitutions of constant length

We consider primitive aperiodic substitutions of constant length q and prove that, in order to have a Lebesgue component in the spectrum of the associated dynamical system, it is necessary that one of the eigenvalues of the substitution matrix equals $\sqrt{q}$ in absolute value. The proof is based on results of M. Queffélec, combined with estimates of the local dimension of the spectral measure at zero.

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arXiv2017

The Hoelder Property for the Spectrum of Translation Flows in Genus Two

The paper is devoted to generic translation flows corresponding to Abelian differentials with one zero of order two on flat surfaces of genus two. These flows are weakly mixing by the Avila-Forni theorem. Our main result gives first quantitative estimates on their spectrum, establishing the Hoelder property for the spectral measures of Lipschitz functions. The proof proceeds via uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markov compacta and arguments of Diophantine nature in the spirit of Salem, Erdos and Kahane.

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arXiv2015

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 parametrized families of self-similar sets and measures in the complex plane, extending earlier results.

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arXiv2014

Connectedness locus for pairs of affine maps and zeros of power series

We study the connectedness locus N for the family of iterated function systems of pairs of affine-linear maps in the plane (the non-self-similar case). First results on the set N were obtained in joint work with P. Shmerkin (2006). Here we establish rigorous bounds for the set N based on the study of power series of special form. We also derive some bounds for the region of "*-transversality" which have applications to the computation of Hausdorff measure of the self-affine attractor. We prove that a large portion of the set N is connected and locally connected, and conjecture that the entire connectedness locus is connected. We also prove that the set N has many zero angle "cusp corners," at certain points with algebraic coordinates.

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arXiv2014

Absolute continuity of self-similar measures, their projections and convolutions

We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small - of co-dimension at least one in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand's projection theorem for planar self-similar sets.

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arXiv2014

On the modulus of continuity for spectral measures in substitution dynamics

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 Hoelder estimate for the spectral measure of almost all suspension flows with a piecewise constant roof function; second, a log-Hoelder estimate for self-similar suspension flows; and, third, a Hoelder asymptotic expansion of the spectral measure at zero for such flows. Our second result implies log-Hoelder 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 the appendix this proposition is used to derive new decay estimates for the Fourier transforms of Bernoulli convolutions.

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arXiv2013

Second Order Ergodic Theorem for Self-Similar Tiling Systems

We consider infinite measure-preserving non-primitive self-similar tiling systems in Euclidean space $\mathbb R^d$. We establish the second-order ergodic theorem for such systems, with exponent equal to the Hausdorff dimension of a graph-directed self-similar set associated with the substitution rule.

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arXiv2012

Dimensions of some fractals defined via the semigroup generated by 2 and 3

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space $Σ_m=\{0,...,m-1\}^\N$ that are invariant under multiplication by integers. The results apply to the sets $\{x\in Σ_m: \forall\, k, \ x_k x_{2k}... x_{n k}=0\}$, where $n\ge 3$. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.

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arXiv2012

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.

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arXiv2012

Dimension spectrum for a nonconventional ergodic average

We compute the dimension spectrum of certain nonconventional averages, namely, the Hausdorff dimension of the set of $0,1$ sequences, for which the frequency of the pattern 11 in positions $k, 2k$ equals a given number $θ\in [0,1]$.

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arXiv2012

The Multiplicative golden mean shift has infinite Hausdorff measure

In an earlier work, joint with R. Kenyon, we computed the Hausdorff dimension of the "multiplicative golden mean shift" defined as the set of all reals in [0,1] whose binary expansion (x_k) satisfies x_k x_{2k}=0 for all k=1,2... Here we show that this set has infinite Hausdorff measure in its dimension. A more precise result in terms of gauges in which the Hausdorff measure is infinite is also obtained.

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arXiv2011

Hausdorff dimension for fractals invariant under the multiplicative integers

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.

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arXiv2011

On the topology of sums in powers of an algebraic number

Let $1<q<2$ and \[ Λ(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $Λ(q)$ is dense in $\mathbb{R}$. We give several sufficient conditions for the denseness of $Λ(q)$ when $q$ is a root of such a polynomial. In particular, we prove that if $q$ is not a Perron number or it has a conjugate $α$ such that $q|α|<1$, then $Λ(q)$ is dense in $\mathbb{R}$.

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