Julien Brémont
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.
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We establish a complete algebraic characterization of self-similar iterated function systems $Φ$ on $\mathbb{R}^{d}$, for which there exists a positive probability vector $p$ so that the Fourier transform of the self-similar measure corresponding to $Φ$ and $p$ does not tend to $0$ at infinity.
Keplerian shear with Rajchman property
本文研究了开普勒剪切在环 bundle 上的特征，利用拉杰恰曼测度分析了动态系统中测度的衰减速度与拉杰恰曼阶数的关系，并扩展到紧致李群 bundle 情况。
On the Rajchman property of homogeneous self-affine measures
本文完整刻画了傅里叶变换不趋于零的齐次自仿测度，证明整数自仿测度具有幂次傅里叶衰减等价于绝对连续，并给出判定绝对连续概率向量的算法。
Random self-similar series over a rotation
本文研究了圆上无理旋转之上的随机自相似级数的律，由此提供了一类自然的连续奇异非Rajchman测度。
Self-similar and self-conformal measures with slow Fourier decay
本文研究了具有慢傅里叶衰减性质的自相似与自共形测度，提出了新的充分条件并构造了具有特殊性质的测度。
Recent progress on pointwise normality of self-similar measures
本文综述自相似测度支撑集中正规数普遍性的最新进展，证明Rajchman性质蕴含典型元素对所有基底正规，改进经典准则，并讨论开放问题。
Let $F$ be a self-similar set on $\mathbb{R}$ associated to contractions $f_j(x) = r_j x + b_j$, $j \in \mathcal{A}$, for some finite $\mathcal{A}$, such that $F$ is not a singleton. We prove that if $\log r_i / \log r_j$ is irrational for some $i \neq j$, then $F$ is a set of multiplicity, that is, trigonometric series are not in general unique in the complement of $F$. No separation conditions are assumed on $F$. We establish our result by showing that every self-similar measure $μ$ on $F$ is a Rajchman measure: the Fourier transform $\widehatμ(ξ) \to 0$ as $|ξ| \to \infty$. The rate of $\wi…
Let $Φ$ be a $C^{1+γ}$ smooth IFS on $\mathbb{R}$, where $γ>0$. We provide mild conditions on the derivative cocycle that ensure that every self conformal measure is supported on points $x$ that are absolutely normal. That is, for integer $p\geq 2$ the sequence $\lbrace p^k x \rbrace_{k\in \mathbb{N}}$ equidistributes modulo $1$. We thus extend several state of the art results of Hochman and Shmerkin about the prevalence of normal numbers in fractals. When $Φ$ is self-similar we show that the set of absolutely normal numbers has full Hausdorff dimension in its attractor, unless $Φ$ has an expl…
Fourier decay and non-decay for pseudo-affine self-conformal measures
研究直线上自共形测度多项式傅里叶衰减充分条件的尖锐性，通过伪仿射迭代函数系统框架，构造非\(C^1\)共轭的迭代函数系统及特定微分同胚，得到相关测度的傅里叶衰减情况。
Commensurating actions and self-similar groups
研究有限生成的收缩自相似分支群的可公度作用并分类，通过建立迭代单值群的FW性质给出无限有限生成顺从群的首个例子，还证明收缩自相似正则分支群不具PW性质，如首个格里戈丘克群。
Liouville property for groups and conformal dimension
本文证明：若迭代单值群的底层空间共形维数小于2，则该群上所有有限二阶矩对称随机游走具有Liouville性质，从而群可均，并应用于次双曲有理函数的Julia集。
Doubling property of self-similar measures with overlaps
本文针对满足特定Pisot数方程的带重叠自相似测度，拓展了已有开集条件下的加倍性研究，证明当参数m≥3时该类测度恒非加倍，补充了m=2时的已知结论。
It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the \emph{weak separation property} is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the \emph{weak separation property} is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use …
We discuss self-similar property of the tricorn, the connectedness locus of the anti-holomorphic quadratic family. As a direct consequence of the study on straightening maps by Kiwi and the author, we show that there are many homeomorphic copies of the Mandelbrot sets. With help of rigorous numerical computation, we also prove that the straightening map is not continuous for the candidate of a "baby tricorn" centered at the airplane, hence not a homeomorphism.
It is known that if the underlying iterated function system satisfies the open set condition, then the upper box dimension of an inhomogeneous self-similar set is the maximum of the upper box dimensions of the homogeneous counterpart and the condensation set. First, we prove that this `expected formula' does not hold in general if there are overlaps in the construction. We demonstrate this via two different types of counterexample: the first is a family of overlapping inhomogeneous self-similar sets based upon Bernoulli convolutions; and the second applies in higher dimensions and makes use of…
Locally measure preserving property of bi-Lipschitz maps between Moran sets
本文证明了在特定类中，两个Moran集之间的双Lipschitz映射同样具有局部测度保持性质，扩展了分形集Lipschitz分类的理论基础。
We introduce a class of self-similar sets which we call {\em twofold Cantor sets} $K_{pq}$ in $\mathbb R$ which are totally disconnected, do not have weak separation property and at the same time have isomorphic self-similar structures.
For a self-similar set in $\mathbb{R}^d$ that is the attractor of an iterated function system that does not verify the weak separation property, Fraser, Henderson, Olson and Robinson showed that its Assouad dimension is at least $1$. In this paper, it is shown that the Assouad dimension of such a set is the sum of the dimension of the vector space spanned by the set of $\textit{overlapping directions}$ and the Assouad dimension of the orthogonal projection of the set self-similar set onto the orthogonal complement of that vector space. This result is applied to give sufficient conditions on th…
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