In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure $μ_λ$ is the probability measure corresponding to the law of the random variable $ξ= \sum_{k=0}^\infty ξ_kλ^k$, where $ξ_k$ are i.i.d. random variables assuming values $-1$ and $1$ with equal probability and $\frac12 < λ< 1$. In particular, for Bernoulli convolutions we give a uniform lower bound $\dim_H(μ_λ) \geq 0.96399$ for all $\frac12<λ<1$.
We show how to turn the question of the absolute continuity of Bernoulli convolutions into one of counting the growth of the number of overlaps in the system. When the contraction parameter is a hyperbolic algebraic integer, we turn this question of absolute continuity into a question involving the ergodic theory of cocycles over domain exchange transformations.
Max-semistable extremal behavior of AR(1)-processes connected with Bernoulli convolutions
Mathematical Institute, Heinrich Heine University Düsseldorf（杜塞尔多夫海因里希·海涅大学数学研究所）；Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen（格罗宁根大学伯努利数学、计算机与人工智能研究所）
Let $λ\in (1,\sqrt{2}]$ be an algebraic integer with Mahler measure $2.$ A classical result of Garsia shows that the Bernoulli convolution $μ_λ$ is absolutely continuous with respect to the Lebesgue measure with a density function in $L^\infty$. In this paper, we show that the density function is continuous.
Spacings and pair correlations for finite Bernoulli convolutions
我们考虑有限伯努利卷积的参数$1/2 < r < 1$支持的离散点集，一般大小为$2^N$。这些序列相对于无限伯努利卷积测度$ν_r$均匀分布，因为$N$趋于无穷大。数值证据表明，对于一个通用的$r$，适当重新缩放点之间的间距分布是泊松分布。我们在这个方向上得到一些部分结果，例如，我们表明，平均而言，对相关性不表现出吸引力或排斥的限制。另一方面，对于某些代数$r$，行为完全不同。
On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions
本文考虑复线性映射对的“Mandelbrot集”$M$，它是由Barnsley和Harrington于1985年提出的，并由Bousch，Bandt等人研究过.它被定义为单位圆盘中的参数$λ$的集合，使得IFS $\{λz-1, λz+1\}$的吸引子$A_λ$是连通的。证明了$M$在虚轴附近的非平凡部分包含在其内部的闭包中（证明了$M$的所有非实点都包含在$M$的内点集的闭包中）.接下来，我们转向吸引子$A_λ$本身以及它们支持的自然度量$ν_λ$。这些措施是复杂的模拟研究无限伯努利卷积。扩展Erdös和Garsia的结果，我们演示了某些类的复杂的代数整数如何产生奇异和绝对连续的措施$ν_λ$。接下来我们研究了$A_λ$的豪斯多夫维数和测度，对于集合$M$中的$λ$，对于Lebesgue-a. e. $λ$。我们还得到了关于a.e.的$ν_λ$的绝对连续性的部分结果。模量大于$\sqrt{1/2}$的$λ$。
We prove that a self similar measure is absolutely continuous providing that it satisfies a condition depending on its Garsia entropy, contraction ratio, and the separation between different points in approximations of the self similar measure. In the special case of Bernoulli convolutions this paper gives a sufficient condition for the Bernoulli convolution with parameter lambda to be absolutely continuous in terms of lambda and its Mahler measure.
For classical Bernoulli convolutions, the Rajchman property, i.e. the convergence to zero at infinity of the Fourier transform, was characterized by successive works of Erd{ö}s [2] and Salem [12]. We prove weak forms of their results for general self-similar measures associated to affine contractions of the real line.
We study the bi-Lipschitz classification of Bedford-McMullen carpets which are totally disconnected. Let $E$ be a such carpet and let $μ_E$ be the uniform Bernoulli measure on $E$. We show that the multifractal spectrum and the doubling property of $μ_E$ are both invariant under a bi-Lipschitz map. Moreover, we show that if $μ_E$ and $μ_F$ are doubling, then a bi-Lipschitz map between $E$ and $F$ enjoys a certain measure preserving property.
