Bernoulli Convolutions Research Articles

On univoque and strongly univoque sets

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet {0,1,…,α} and a real number (base) 1<β<α+1, the so-called univoque set of numbers which have a unique expansion in base β has garnered a great deal of attention in recent years. Motivated by recent applications of β-expansions to Bernoulli convolutions and a certain class of self-affine functions, we introduce the notion of a strongly univoque set. We study in detail the set Dβ of numbers which are univoque but not strongly univoque. Our main result is that Dβ is nonempty if and only if the number 1 has a unique nonterminating expansion in base β, and in that case, Dβ is uncountable. We give a sufficient condition for Dβ to have positive Hausdorff dimension, and show that, on the other hand, there are infinitely many values of β for which Dβ is uncountable but of Hausdorff dimension zero.

Polynomial approximation of self-similar measures and the spectrum of the transfer operator

We consider self-similar measures on \begin{document} $\mathbb{R}.$ \end{document} The Hutchinson operator \begin{document} $H$ \end{document} acts on measures and is the dual of the transfer operator \begin{document} $T$ \end{document} which acts on continuous functions. We determine polynomial eigenfunctions of \begin{document} $T.$ \end{document} As a consequence, we obtain eigenvalues of \begin{document} $H$ \end{document} and natural polynomial approximations of the self-similar measure. Bernoulli convolutions are studied as an example.

Local dimensions of measures of finite type

We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar measures associated with m -fold sums of Cantor sets with ratio of dissection 1/R for integer R \leq m . We introduce a combinatorial notion called a loop class and prove that the set of attainable local dimensions of the measure at points in a positive loop class is a closed interval. We prove that the local dimensions at the periodic points in the loop class are dense and give a simple formula for those local dimensions. These self-similar measures have a distinguished positive loop class called the essential class. The set of points in the essential class has full Lebesgue measure in the support of the measure and is often all but the two endpoints of the support. Thus many, but not all, measures of finite type have at most one isolated point in their set of local dimensions. We give examples of Bernoulli convolutions whose sets of attainable local dimensions consist of an interval together with an isolated point. As well, we give an example of a measure of finite type that has exactly two distinct local dimensions.

Sofic Measures and Densities of Level Sets

Abstract The Bernoulli convolution associated to the real β &gt; 1 and the probability vector (p 0, . . . , pd −1) is a probability measure ηβ,p on ℝ, solution of the self-similarity relation η = ∑ k = 0 d − 1 p k ⋅ η ○ S k − 1 $\eta = \sum\nolimits_{k = 0}^{d - 1} {p_k \cdot \eta \circ S_k^{ - 1} } $ , where S k ( x ) = x + k β $S_k (x) = {{x + k} \over \beta }$ . If β is an integer or a Pisot algebraic number with finite Rényi expansion, ηβ,p is sofic and a Markov chain is naturally associated. If β = b ∈ ℕ and p 0 = ⋯ = p d − 1 = 1 d $p_0 = \cdots = p_{d - 1} = {1 \over d}$ , the study of ηb,p is close to the study of the order of growth of the number of representations in base b with digits in {0, 1, . . . , d − 1}. In the case b = 2 and d = 3 it has also something to do with the metric properties of the continued fractions.

Variability and singularity arising from poor compliance in a pharmacokinetic model II: the multi-oral case.

We propose a stochastic model for the drug concentration in the case of multiple oral doses and in a situation of poor patient adherence. Our model is able to take into account an irregular drug intake schedule. This article is the second in a series of three. It presents a multi-oral version of the results given in Lévy-Véhel and Lévy-Véhel (J Pharmacokinet Pharmacodyn 40(1):15-39, 2013), that dealt with the multi-IV bolus case. Under the assumption that the irregular dosing schedule follows a Poisson law, we study features of the drug concentration that have practical implications, such as its variability and the regularity of its cumulative probability distribution, which describes its predictive power with respect to the mean behaviour. We consider four variants: continuous-time, with either deterministic or random doses, and discrete-time, also with either deterministic or random doses. Our computations allow one to assess in a precise way the effect of various significant parameters such as the mean rate of intake, the elimination rate, the absorption rate and the mean dose. They quantify how much poor adherence will affect the efficacy of therapy. To appreciate this impact, we provide detailed comparisons with the variability of concentration in two reference situations: a fully adherent patient and a population of fully adherent patients with log-normally distributed pharmacokinetic parameters. Besides, the discrete-time versions of our models reveal unexpected links with objects which have been studied in the mathematical literature under the name of infinite Bernoulli convolutions (Erdós, Am J Math 61:974-975, 1939). This allows us to quantify the fact that, when the random dosing schedule is too sparse, the concentration behaves in a very erratic way. Our results complement the ones in Lévy-Véhel and Lévy-Véhel (J Pharmacokinet Pharmacodyn 40(1):15-39, 2013) and help understanding the consequences of poor adherence. They may have practical outcomes in terms of drug dosing and scheduling.

Absolute continuity of complex Bernoulli convolutions

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

Multiple Spectra of Bernoulli Convolutions

AbstractLet μλ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ1/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μλ shows that μλ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ1/(2k). This is surprising because some spectra for the measure μ1/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ1/(2k).

Bernoulli and poly-Bernoulli polynomial convolutions and identities of p-adic Arakawa–Kaneko zeta functions

We evaluate the ordinary convolution of Bernoulli polynomials in closed form in terms of poly-Bernoulli polynomials. As applications we derive identities for [Formula: see text]-adic Arakawa–Kaneko zeta functions, including a [Formula: see text]-adic analogue of Ohno’s sum formula. These [Formula: see text]-adic identities serve to illustrate the relationships between real periods and their [Formula: see text]-adic analogues.

Singularity and Fine Fractal Properties of One Class of Infinite Bernoulli Convolutions with Essential Overlapping. II

We study the Lebesgue structure and fine fractal properties of infinite Bernoulli convolutions, i.e., the distributions of random variables of the form ξ = ∑ = 1 ∞ ξ k a k , where ∑ = 1 ∞ a k is a convergent positive series and ξ k are independent (generally speaking, nonidentically distributed) Bernoulli random variables. Our main aim is to investigate the class of Bernoulli convolutions with essential overlaps generated by a series ∑ = 1 ∞ a k such that, for any k ∈ ℕ, there exists s k ∈ ℕ ⋃{0} for which a k = a k+1 =…= $$ {a}_{k+{s}_k} $$ ≥ $$ {r}_{k+{s}_k} $$ and, in addition, s k > 0 for infinitely many indices k. In this case, almost all (both in a sense of Lebesgue measure and in a sense of fractal dimension) points of the spectrum have continuum many representations of the form ξ = ∑ = 1 ∞ e k a k with e k ∈ {0, 1}. It is shown that the probability measure μ ξ has either a pure discrete distribution or a pure singularly continuous distribution. We also establish sufficient conditions for the confidentiality of the family of cylindrical intervals on the spectrum $$ {S}_{\mu_{\xi }} $$ generated by the distributions of the random variable ξ. In the case of singularity, we also deduce the explicit formula for the Hausdorff dimension of the corresponding probability measure [i.e., the Hausdorff–Besicovitch dimension of the minimal supports of the measure μ ξ (in a sense of dimension)].

Asymptotic formula for the moments of Bernoulli convolutions

. Asymptotic Formula for the Moments of Bernoulli Convolutions Timofeev E. A. Received February 8, 2016 For each λ , 0 &lt; λ &lt; 1 , we define a random variable ∞ Y λ =(1 − λ ) ξ n λ n , n =0 where ξ n are independent random variables with P { ξ n =0 } = P { ξ n =1 } = 1 . 2 The distribution of Y λ is called a symmetric Bernoulli convolution. The main result of this paper is M n = E Y λ n = n log λ 2 2 log λ (1 − λ )+0 . 5log λ 2 − 0 . 5 e τ ( − log λ n ) 1+ O ( n − 0 . 99 ) , where is a 1-periodic function, 1 k 2 πikx τ ( x )= k α − ln λ e k ̸ =0 1 (1 − λ ) 2 πit (1 − 2 2 πit ) π − 2 πit 2 − 2 πit ζ (2 πit ) , 2 i sh( π 2 t ) α ( t ) = − and ζ ( z ) is the Riemann zeta function. The article is published in the author’s wording.

Sets of $\beta$-expansions and the Hausdorff measure of slices through fractals

We study natural measures on sets of \beta -expansions and on slices through self similar sets. In the setting of \beta -expansions, these allow us to better understand the measure of maximal entropy for the random \beta -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Hausdorff measure, generalising long known results about almost everywhere values of the Hausdorff dimension.

Random walks in the group of Euclidean isometries and self-similar measures

We study products of random isometries acting on Euclidean space. Building on previous work of the second author, we prove a local limit theorem for balls of shrinking radius with exponential speed under the assumption that a Markov operator associated to the rotation component of the isometries has spectral gap. We also prove that certain self-similar measures are absolutely continuous with smooth densities. These families of self-similar measures give higher dimensional analogues of Bernoulli convolutions on which absolute continuity can be established for contraction ratios in an open set.

Overlap Functions for Measures in Conformal Iterated Function Systems

We study conformal iterated function systems (IFS) $\mathcal S = \{\phi_i\}_{i \in I}$ with arbitrary overlaps, and measures $\mu$ on limit sets $\Lambda$, which are projections of equilibrium measures $\hat \mu$ with respect to a certain lift map $\Phi$ on $\Sigma_I^+ \times \Lambda$. No type of Open Set Condition is assumed. We introduce a notion of overlap function and overlap number for such a measure $\hat \mu$ with respect to $\mathcal S$; and, in particular a notion of (topological) overlap number $o(\mathcal S)$. These notions take in consideration the $n$-chains between points in the limit set. We prove that $o(\mathcal S, \hat \mu)$ is related to a conditional entropy of $\hat \mu$ with respect to the lift $\Phi$. Various types of projections to $\Lambda$ of invariant measures are studied. We obtain upper estimates for the Hausdorff dimension $HD(\mu)$ of $\mu$ on $\Lambda$, by using pressure functions and $o(\mathcal S, \hat \mu)$. In particular, this applies to projections of Bernoulli measures on $\Sigma_I^+$. Next, we apply the results to Bernoulli convolutions $\nu_\lambda$ for $\lambda \in (\frac 12, 1)$, which correspond to self-similar measures determined by composing, with equal probabilities, the contractions of an IFS with overlaps $\mathcal S_\lambda$. We prove that for all $\lambda \in (\frac 12, 1)$, there exists a relation between $HD(\nu_\lambda)$ and the overlap number $o(\mathcal S_\lambda)$. The number $o(\mathcal S_\lambda)$ is approximated with integrals on $\Sigma_2^+$ with respect to the uniform Bernoulli measure $\nu_{(\frac 12, \frac 12)}$. We also estimate $o(\mathcal S_\lambda)$ for certain values of $\lambda$.

Bernoulli convolutions and 1D dynamics

We describe a family of dynamical systems on the unit interval which preserve Bernoulli convolutions. We show that if there are parameter ranges for which these systems are piecewise convex, then the corresponding Bernoulli convolution will be absolutely continuous with bounded density. We study the systems and give some numerical evidence to suggest values of λ for which may be piecewise convex.

Discrete random walks on the group Sol

The harmonic measure $\nu$ on the boundary of the group $Sol$ associated to a discrete random walk of law $\mu$ was described by Kaimanovich. We investigate when it is absolutely continuous or singular with respect to Lebesgue measure. By ratio entropy over speed, we show that any countable non-abelian subgroup admits a finite first moment non-degenerate $\mu$ with singular harmonic measure $\nu$. On the other hand, we prove that some random walks with finitely supported step distribution admit a regular harmonic measure. Finally, we construct some exceptional random walks with arbitrarily small speed but singular harmonic measures. The two later results are obtained by comparison with Bernoulli convolutions, using results of Erd\"os and Solomyak.

One-dimensional wave equations defined by fractal Laplacians

We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the post-critically finite condition or the open set condition. By using second-order self-similar identities introduced by Strichartz et al., we discretize the equations and use the finite element and central difference methods to obtain numerical approximations to the weak solutions. We prove that the numerical solutions converge to the weak solution, and obtain estimates for the rate of convergence.

Sierpinski-type spectral self-similar measures

We consider the Sierpinski-type self-similar measure μ ρ on R 2 with contraction ratio 0 < | ρ | < 1 , we show that μ ρ is a spectral measure if and only if | ρ | = 1 / ( 3 p ) for some integer p > 0 . A similar characterization for Bernoulli convolution is due to Dai [2] , over which ρ = 1 / ( 2 p ) .

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.

Spectrality of infinite Bernoulli convolutions

Let { b k − a k } k = 1 ∞ be a sequence of positive integers with upper bound and let δ E be the uniformly discrete probability measure on the finite set E . For 0 < ρ < 1 , the infinite convolution μ ρ , { a k , b k } = δ ρ { a 1 , b 1 } ⁎ δ ρ 2 { a 2 , b 2 } ⁎ ⋯ ⋯ is called an infinite Bernoulli convolution. In this paper, we investigate whenever there exists Λ such that { e − 2 π i λ x } λ ∈ Λ is an orthonormal basis for L 2 ( μ ρ , { a k , b k } ) .

On self-similar sets with overlaps and inverse theorems for entropy

We study the Hausdorff dimension of self-similar sets and measures on the line. We show that if the dimension is smaller than the minimum of 1 and the similarity dimension, then at small scales there are super-exponentially close cylinders. This is a step towards the folklore conjecture that such a drop in dimension is explained only by exact overlaps, and confirms the conjecture in cases where the contraction parameters are algebraic. It also gives an affirmative answer to a conjecture of Furstenberg, showing that the projections of the "1-dimensional Sierpinski gasket" in irrational directions are all of dimension 1. As another consequence, if a family of self-similar sets or measures is parametrized in a real-analytic manner, then, under an extremely mild non-degeneracy condition, the set of "exceptional" parameters has Hausdorff dimension 0. Thus, for example, there is at most a zero-dimensional set of parameters 1/2<r<1 such that the corresponding Bernoulli convolution has dimension <1, and similarly for Sinai's problem on iterated function systems that contract on average. A central ingredient of the proof is an inverse theorem for the growth of Shannon entropy of convolutions of probability measures. For the dyadic partition D_n of the line into intervals of length 1/2^n, we show that if H(nu*mu,D_n)/n < H(mu,D_n)/n + delta for small delta and large n, then, when restricted to random element of a partition D_i, 0<i<n, either mu is close to uniform or nu is close to atomic. This should be compared to results in additive combinatorics that give the global structure of measures satisfying H(nu*mu,D_n)/n < H(mu,D_n)/n + O(1/n).