Phase transitions for the multifractal analysis of self-similar measures

On multifractal formalism for self-similar measures with overlaps

Let $μ$ be a self-similar measure generated by an IFS $Φ=\{ϕ_i\}_{i=1}^\ell$ of similarities on $\mathbb R^d$ ($d\ge 1$). When $Φ$ is dimensional regular (see Definition~1.1), we give an explicit formula for the $L^q$-spectrum $τ_μ(q)$ of $μ$ over $[0,1]$, and show that $τ_μ$ is differentiable over $(0,1]$ and the multifractal formalism holds for $μ$ at any $α\in [τ_μ'(1),τ_μ'(0+)]$. We also verify the validity of the multifractal formalism of $μ$ over $[τ_μ'(\infty),τ_μ'(0+)]$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds.…

The pointwise Hölder spectrum of general self-affine functions on an interval

This paper gives the pointwise Hölder (or multifractal) spectrum of continuous functions on the interval $[0,1]$ whose graph is the attractor of an iterated function system consisting of $r\geq 2$ affine maps on $\mathbb{R}^2$. These functions satisfy a functional equation of the form $ϕ(a_k x+b_k)=c_k x+d_kϕ(x)+e_k$, for $k=1,2,\dots,r$ and $x\in[0,1]$. They include the Takagi function, the Riesz-Nagy singular functions, Okamoto's functions, and many other well-known examples. It is shown that the multifractal spectrum of $ϕ$ is given by the multifractal formalism when $|d_k|\geq |a_k|$ for a…

自相似多重分形测度的几何与组合性质

Geometric and Combinatorial Properties of Self-similar Multifractal Measures

本文通过构造有向图控制自相似测度的分布，推广净区间方法，在弱分离及广义有限类型条件下验证多重分形形式体系，并应用于具有精确重叠的不可数IFS族。

在缺项小波级数的分形支撑内构建自相似子集以进行其多重分形分析

Constructing self-similar subsets within the fractal support of Lacunary Wavelet Series for their multifractal analysis

针对Hausdorff维数与上盒维数匹配的分形，提出在内部构建具有可控自相似性的拟Cantor子集的新方法，用于在经典质量传递原理失效时估计相交集合的维数下界，并应用于计算缺项小波级数的递增多重分形谱。

Multifractal analysis of Bernoulli convolutions associated with Salem numbers

We consider the multifractal structure of the Bernoulli convolution $ν_λ$, where $λ^{-1}$ is a Salem number in $(1,2)$. Let $τ(q)$ denote the $L^q$ spectrum of $ν_λ$. We show that if $α\in [τ'(+\infty), τ'(0+)]$, then the level set $$E(α):={x\in \R:\; \lim_{r\to 0}\frac{\log ν_λ([x-r, x+r])}{\log r}=α}$$ is non-empty and $\dim_HE(α)=τ^*(α)$, where $τ^*$ denotes the Legendre transform of $τ$. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval $[τ'(+\infty), τ'(0+)]$ is not a singleton when $λ^{-1}$ is the la…

参数化von Koch函数族的多重分形性质

The multifractal nature of a parametrized family of von Koch functions

本文证明参数化von Koch函数族具有丰富的多重分形行为，通过研究底层动力系统轨道并引入自相似测度与非平凡迭代函数系统，推广了经典von Koch曲线的正则性性质。

通用投影定理及其在多分形分析中的应用与自共形集上每个遍历测度维数同时成立的结果

Universal projection theorems with applications to multifractal analysis and the dimension of every ergodic measure on self-conformal sets simultaneously

本文提出通用投影定理，在横截性和相对维数可分条件下证明投影维数等式及近似双 Lipschitz 性质对一族测度同时成立，并应用于多分形分析、遍历测度和 Gibbs 测度的维数公式。