scholarly journals When does a Bernoulli convolution admit a spectrum?

2012 ◽  
Vol 231 (3-4) ◽  
pp. 1681-1693 ◽  
Author(s):  
Xin-Rong Dai
Keyword(s):  
Bernoulli Convolution

On the spectrality of self-affine measures with four digits on ℝ2

2021 ◽  
pp. 2150004
Author(s):  
Ming-Liang Chen ◽  
Zhi-Hui Yan
Keyword(s):  
Spectral Property ◽  
Orthonormal Basis ◽  
Zero Set ◽  
Real Matrix ◽  
Exponential Functions ◽  
Bernoulli Convolution ◽  
Digit Set ◽  
Similar Characterization ◽  
Equivalent Characterization ◽  
Matrix Formula

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

scholarly journals On the Hausdorff Dimension of Bernoulli Convolutions

2018 ◽  
Vol 2020 (19) ◽  
pp. 6569-6595 ◽  
Author(s):  
Shigeki Akiyama ◽  
De-Jun Feng ◽  
Tom Kempton ◽  
Tomas Persson
Keyword(s):  
Hausdorff Dimension ◽  
Rate Of Convergence ◽  
Finite Time ◽  
Time Algorithm ◽  
Bernoulli Convolution ◽  
Bernoulli Convolutions ◽  
Explicit Rate ◽  
Products Of Matrices

Abstract We give an expression for the Garsia entropy of Bernoulli convolutions in terms of products of matrices. This gives an explicit rate of convergence of the Garsia entropy and shows that one can calculate the Hausdorff dimension of the Bernoulli convolution $\nu _{\beta }$ to arbitrary given accuracy whenever $\beta $ is algebraic. In particular, if the Garsia entropy $H(\beta )$ is not equal to $\log (\beta )$ then we have a finite time algorithm to determine whether or not $\operatorname{dim_H} (\nu _{\beta })=1$.

scholarly journals Local Dimensions of Measures of Finite Type II: Measures Without Full Support and With Non-regular Probabilities

2018 ◽  
Vol 70 (4) ◽  
pp. 824-867 ◽  
Author(s):  
Kathryn E. Hare ◽  
Kevin G. Hare ◽  
Michael Ka Shing Ng
Keyword(s):  
Finite Type ◽  
Hausdorff Measure ◽  
Finite Sequence ◽  
Pisot Number ◽  
Local Dimension ◽  
Bernoulli Convolution ◽  
Similar Measure ◽  
Absolutely Continuous ◽  
Contraction Ratio ◽  
Isolated Point

AbstractConsider a finite sequence of linear contractions Sj(x) = px + dj and probabilities pj > 0 with ∑Pj = 1. We are interested in the self-similar measure , of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.Under some mild technical assumptions, we prove that there exists a subset of supp μ of full μ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support, and we show that the dimension of the support can be computed using only information about the essential class.To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the k-th convolution of the associated Cantor measure has local dimension at x ∊ (0,1) tending to 1 as ft: tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.

Spectrality of Moran measures with finite arithmetic digit sets

2019 ◽  
Vol 31 (02) ◽  
pp. 2050008
Author(s):  
Zong-Sheng Liu ◽  
Xin-Han Dong
Keyword(s):  
Probability Measure ◽  
Borel Probability Measure ◽  
Bernoulli Convolution ◽  
Equal Distribution ◽  
Finite Set ◽  
Convolution Formula ◽  
Borel Probability ◽  
Finite Arithmetic ◽  
Moran Measure ◽  
Uniformly Bounded

Let [Formula: see text] be a prime and [Formula: see text] be a sequence of finite arithmetic digit sets in [Formula: see text] with [Formula: see text] uniformly bounded, and let [Formula: see text] be the discrete probability measure on the finite set [Formula: see text] with equal distribution. For [Formula: see text], the infinite Bernoulli convolution [Formula: see text] converges weakly to a Borel probability measure (Moran measure). In this paper, we study the existence of exponential orthonormal basis for [Formula: see text].

Multiple Spectra of Bernoulli Convolutions

2016 ◽  
Vol 60 (1) ◽  
pp. 187-202 ◽  
Author(s):  
Jian-Lin Li ◽  
Dan Xing
Keyword(s):  
Finite Group ◽  
Spectral Measure ◽  
Bernoulli Convolution ◽  
Bernoulli Convolutions ◽  
Order Of Elements

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).

Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps

2011 ◽  
Vol 63 (3) ◽  
pp. 648-688 ◽  
Author(s):  
Sze-Man Ngai
Keyword(s):  
Golden Ratio ◽  
Iterated Function Systems ◽  
Spectral Dimension ◽  
Open Set Condition ◽  
Bernoulli Convolution ◽  
One Dimensional ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Set Up

AbstractWe set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

scholarly journals LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

2020 ◽  
pp. 15-18
Author(s):  
O. Makarchuk ◽  
K. Salnik
Keyword(s):  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Special Kind ◽  
Random Variable ◽  
Random Series ◽  
Bernoulli Convolution ◽  
Bernoulli Convolutions ◽  
Lévy’S Theorem ◽  
Distribution Matrix ◽  
Necessary And Sufficient

The paper deals with the problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions of a special kind. The main attention is paid to the case when the terms of a random series acquire three values: 0, 1, 2. In the case when the probability that the term of a random series becomes 2 is 0, the corresponding generalized Bernoulli convolutions coincide with classic Bernoulli convolutions, which were actively studied domestic scientists (Pratsovyty M., Turbin G., Torbin G., Honcharenko Ya., Baranovsky O., Savchenko I. and others) as well as foreign researchers (Erdos P., Peres Y., Schlag W, Solomyak B., Albeverio S. and others). The problem of deepening the Jessen-Wintner theorem concerning the necessary and sufficient conditions for the distribution of a probably convergent random series with discrete additions to each of the three pure types, is extremely difficult to formulate and is not completely solved even for classical Bernoulli convolutions. The results of the study are a deepening in relation to the analysis of the Lebesgue structure of random series formed by s-expansions of real numbers. In the case when the corresponding Bernoulli convolution is generated by the sequence 3-n, we have a random variable with independent triple digits, which was studied by scientists in different directions: Lebesgue structure (Chaterji S., Marsaglia G.), topological-metric structure of the distribution spectrum (Pratsovityi M., Turbin G.), fractal analysis of the distribution carrier (Pratsovyty M., Torbin G.), asymptotic properties of the characteristic function at infinity (Honcharenko Ya., Pratsovyty M., Torbin G.). The paper presents certain sufficient conditions for the absolute continuity and singularity of the distribution, with certain restrictions on the stochastic distribution matrix and the asymptotics of the values of the random terms of the series. In the case when the Lebesgue measure of the set of realizations of the generalized Bernoulli convolution is different from zero, it is possible together with Levy's theorem to formulate criteria for belonging of the Bernoulli convolution distribution to each of the three pure Lebesgue types, namely: purely discrete, purely continuous or purely singular.

scholarly journals A lower bound for Garsia’s entropy for certain Bernoulli convolutions

2010 ◽  
Vol 13 ◽  
pp. 130-143 ◽  
Author(s):  
Kevin G. Hare ◽  
Nikita Sidorov
Keyword(s):  
Lower Bound ◽  
High Precision ◽  
Pisot Number ◽  
Bernoulli Convolution ◽  
Pisot Numbers ◽  
Bernoulli Convolutions

AbstractLetβ(1,2) be a Pisot number and letHβdenote Garsia’s entropy for the Bernoulli convolution associated withβ. Garsia, in 1963, showed thatHβ<1 for any Pisotβ. For the Pisot numbers which satisfyxm=xm−1+xm−2++x+1 (withm≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier form=2 and later by Grabner, Kirschenhofer and Tichy form≥3, and it proves to be close to 1. No other numerical values forHβare known. In the present paper we show thatHβ>0.81 for all Pisotβ, and improve this lower bound for certain ranges ofβ. Our method is computational in nature.

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