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Cai-Yun Ma

Publications and source records attributed to Cai-Yun Ma.

3 recordsLinked to original sources
arXiv2026

Separation Conditions for Iterated Function Systems with a Common Attractor

Let $K$ be the non-singleton attractor of a finite iterated function system $Φ=\{ϕ_i(x)=a_i O_i x+t_i\}_{i=1}^m$ on $\mathbb{R}^d$, where $0<a_i<1$, $t_i\in\mathbb{R}^d$ and the $O_i$ are orthogonal transformations on $\mathbb{R}^d$. Suppose that $Φ$ satisfies the open set condition, but not the strong separation condition. We show that $K$ cannot be generated by any finite iterated function system of similitudes satisfying the strong separation condition. This removes the homogeneity assumption from a result of Feng, Ruan and Xiong and gives a negative answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.

math.DS↗
arXiv2023

Dimensions of projected sets and measures on typical self-affine sets

Let $T_1,\ldots, T_m$ be a family of $d\times d$ invertible real matrices with $\|T_i\|<1/2$ for $1\leq i\leq m$. For ${\bf a}=(a_1,\ldots, a_m)\in \Bbb R^{md}$, let $π^{\bf a}:\; Σ=\{1,\ldots, m\}^{\Bbb N}\to \Bbb R^d$ denote the coding map associated with the affine IFS $\{T_ix+a_i\}_{i=1}^m$. We show that for every Borel probability measure $μ$ on $Σ$, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of $π^{\bf a}_*μ$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$, where $π^{\bf a}_*μ$ stands for the push-forward of $μ$ by $π^{\bf a}$. In particular, we give a necessary and sufficient condition on $μ$ so that $π^{\bf a}_*μ$ is exact dimensional for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Moreover, for every analytic set $E\subset Σ$, each of the Hausdorff, packing, lower and upper box-counting dimensions of $π^{\bf a}(E)$ is constant for $\mathcal L^{md}$-a.e.~${\bf a}\in \Bbb R^{md}$. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.

math.DS↗