arXiv subjects
Publications and source records attributed to Boris Solomyak.
We compute 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 $k\ge 1$, and show that it is smaller than the Minkowski dimension.
We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces. It is shown that every ergodic invariant measure (finite or "regular" infinite) is obtained by an extension from a simple subdiagram. We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. A number of sufficient conditions for unique ergodicity are obtained. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. On the other hand, for the so-called "consecutive" ordering, the Vershik map is not strongly mixing on all finite rank diagrams.
Let $ν_λ^p$ be the distribution of the random series $\sum_{n=1}^\infty i_n λ^n$, 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 $ν_λ^p$ for typical $λ$. Namely, we investigate the size of the sets \[ Δ_{λ,p}(α) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log ν_λ^p(B(x,r))}{\log r} =α\right\}. \] Our main results highlight the fact that for almost all, and in some cases all, $λ$ in an appropriate range, $Δ_{λ,p}(α)$ is nonempty and, moreover, has positive Hausdorff dimension, for many values of $α$. This happens even in parameter regions for which $ν_λ^p$ is typically absolutely continuous.
We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well-known that in the primitive case the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive, and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize $σ$-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the "integer Sierpiński gasket and carpet" tilings. For such tilings the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported $σ$-finite invariant measure, which is locally finite and unique up to scaling.
We consider self-affine tilings in $\R^n$ with expansion matrix $ϕ$ and address the question which matrices $ϕ$ can arise this way. In one dimension, $λ$ is an expansion factor of a self-affine tiling if and only if $|λ|$ is a Perron number, by a result of Lind. In two dimensions, when $ϕ$ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex $λ$ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for $ϕ$ to be an expansion matrix for any $n$, assuming only that $ϕ$ is diagonalizable over the complex numbers. We conjecture that this condition on $ϕ$ is also sufficient for the existence of a self-affine tiling.
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. It turns out that the presence of the discrete component depends on the algebraic properties of the eigenvalues of the expansion matrix $ϕ$ for the tiling. Assuming that $ϕ$ is diagonalizable over $\C$ and all its eigenvalues are algebraic conjugates of the same multiplicity, we show that the dynamical system has a relatively dense discrete spectrum if and only if it is not weakly mixing, and if and only if the spectrum of $ϕ$ is a "Pisot family". Moreover, this is equivalent to the Meyer property of the associated discrete set of "control points" for the tiling.
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.
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.
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. We prove that if $A$ has bounded turning and $p$ has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS $\{λz,λz+1\}$ where $λ$ is a complex parameter in the unit disk, such that its attractor $A_\lam$ is a dendrite, which happens whenever $\Ok$ is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map $q_\lam$ on $A_\lam$. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map $p_c(z)=z^2+c$, with the Julia set $J_c$ such that $(A_\lam,q_\lam)$ and $(J_c,p_c)$ are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.
We consider finite Bernoulli convolutions with a parameter $1/2 < r < 1$ supported on a discrete point set, generically of size $2^N$. These sequences are uniformly distributed with respect to the infinite Bernoulli convolution measure $ν_r$, as $N$ tends to infinity. Numerical evidence suggests that for a generic $r$, the distribution of spacings between appropriately rescaled points is Poissonian. We obtain some partial results in this direction; for instance, we show that, on average, the pair correlations do not exhibit attraction or repulsion in the limit. On the other hand, for certain algebraic $r$ the behavior is totally different.
We prove that for the uniquely ergodic ${\bf R}^d$ action associated with a primitive substitution tiling of finite local complexity, every measurable eigenfunction coincides with a continuous function almost everywhere. Thus, topological weak-mixing is equivalent to measure-theoretic weak-mixing for such actions. If the expansion map for the substitution is a pure dilation by $θ>1$ and the substitution has a fixed point, then failure of weak-mixing is equivalent to $θ$ being a Pisot number.
We consider a Branching Random Walk on $\R$ whose step size decreases by a fixed factor, $0 1/2$ the limit measure is almost surely (a.s.) absolutely continuous with respect to the Lebesgue measure, but for Pisot $1/b$ it is a.s. singular; (2) for all $b > (\sqrt{5}-1)/2$ the support of the measure is a.s. the closure of its interior; (3) for Pisot $1/b$ the support of the measure is ``fractured'': it is a.s. disconnected and the components of the complement are not isolated on both sides.
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.
We prove that a purely unrectifiable self-similar set of finite 1-dimensional Hausdorff measure in the plane, satisfying the Open Set Condition, has radial projection of zero length from every point.
It is proved that every pseudo-self-affine tiling in R^d is mutually locally derivable with a self-affine tiling. A characterization of pseudo-self-similar tilings in terms of derived Voronoi 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.
We investigate topological mixing for Z and R actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $θ_2$ of the substitution matrix satisfies $|θ_2|\ne 1$. If $|θ_2|<1$, then (as is well-known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|θ_2|> 1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|θ_2|=1$ is more delicate, and we only obtain some partial results.
We consider linear iterated function systems with a random multiplicative error on the real line. Our system is $\{x\mapsto d_i + λ_i Y x\}_{i=1}^m$, where $d_i\in \R$ and $λ_i>0$ are fixed and $Y> 0$ is a random variable with an absolutely continuous distribution. The iterated maps are applied randomly according to a stationary ergodic process, with the sequence of i.i.d. errors $y_1,y_2,...$, distributed as $Y$, independent of everything else. Let $h$ be the entropy of the process, and let $χ= E[\log(λY)]$ be the Lyapunov exponent. Assuming that $χ< 0$, we obtain a family of conditional measures $ν_y$ on the line, parametrized by $y = (y_1,y_2,...)$, the sequence of errors. Our main result is that if $h > |χ|$, then $ν_y$ is absolutely continuous with respect to the Lebesgue measure for a.e. $y$. We also prove that if $h < |χ|$, then the measure $ν_y$ is singular and has dimension $h/|χ|$ for a.e. $y$. These results are applied to a randomly perturbed IFS suggested by Y. Sinai, and to a class of random sets considered by R. Arratia, motivated by probabilistic number theory.
In a recent paper, Pertti Mattila asked which gauge functions $ϕ$ have the property that for any planar Borel set $A$ with positive Hausdorff measure in gauge $ϕ$, the projection of $A$ to almost every line has positive length. We show that integrability near zero of $ϕ(r)/(r^2)$, which is known to be sufficient for this property, is also necessary if $ϕ$ is regularly varying. Our proof is based on a random construction adapted to the gauge function.