无无理性假设下自相似集的笛卡儿积的投影
Projections of cartesian products of the self-similar sets without the irrationality assumption
摘要：
让$β>1$。定义一类自相似集\[S=\left\{f_{i}(x)=\dfrac{x}{β^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\]设$\mathcal{A}$为自相似集从$S$生成的所有自相似集的集合。本文证明了对任意$θ\in(0,π)$和$K_1, K_2\in \mathcal{A}$，$Proj_θ(K_1\times K_2)$类似于某个无穷迭代函数系的自相似集或吸引子，其中$Proj_θ$表示在$L_θ$上的正交投影，$L_θ$表示在$θ$方向上通过原点的直线.作为推论，$\dim_{P}(Proj_θ(K_1\times K_2))=\overline{\dim}_{B}(Proj_θ(K_1\times K_2))$适用于任何$θ\in(0,π)$和任何$K_1, K_2\in \mathcal{A}$，其中$\dim_{P}$和$\overline{\dim}_{B}$表示填充和上盒维数。$Proj_θ(K_1\times K_2)$是否相似于自相似集，是由$K_1$和$K_2$的相似比唯一决定的，而不是由$θ.$角唯一决定的。当$Proj_θ(K_1\times K_2)$相似于自相似集时，根据有限型条件\cite{NW}，我们可以在一定情况下计算出$Proj_θ(K_1\times K_2)$的Hausdorff维数。如果$Proj_θ(K_1\times K_2)$类似于某个无穷迭代函数系统的吸引子，则利用Vitali覆盖引理\cite{FG}给出了$Proj_θ(K_1\times K_2)$的Hausdorff维数的估计.在某些情况下，利用Mauldin和Urbanski的结果\cite{MRD}，我们可以计算出$Proj_θ(K_1\times K_2)$的精确Hausdorff维数.我们还发现了一些非平凡的例子，使得对于一些角度$θ\in[0,π)$和一些$K_1, K_2\in \mathcal{A}$，$\dim_{H}(Proj_θ(K_1\times K_2))=\dim_{H}(K_1)+\dim_{H}(K_2)$。
英文摘要：
Let $β>1$. Define a class of similitudes \[S=\left\{f_{i}(x)=\dfrac{x}{β^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\] Let $\mathcal{A}$ be the collection of all the self-similar sets generated by the similitudes from $S$. In this paper, we prove that for any $θ\in(0,π)$ and $K_1, K_2\in \mathcal{A}$, $Proj_θ(K_1\times K_2)$ is similar to a self-similar set or an attractor of some infinite iterated function system, where $Proj_θ$ denotes the orthogonal projection onto $L_θ$, and $L_θ$ denotes the line through the origin in direction $θ$. As a corollary, $\dim_{P}(Proj_θ(K_1\times K_2))=\overline{\dim}_{B}(Proj_θ(K_1\times K_2))$ holds for any $θ\in(0,π)$ and any $K_1, K_2\in \mathcal{A}$, where $\dim_{P}$ and $\overline{\dim}_{B}$ denote the packing and upper box dimension. Whether $Proj_θ(K_1\times K_2)$ is similar to a self-similar set or not is uniquely determined by the similarity ratios of $K_1$ and $K_2$ rather than the angle $θ.$ When $Proj_θ(K_1\times K_2)$ is similar to a self-similar set, in terms of the finite type condition \cite{NW}, we are able to calculate in cerntain cases the Hausdorff dimension of $Proj_θ(K_1\times K_2)$. If $Proj_θ(K_1\times K_2)$ is similar to an attractor of some infinite iterated function system, then by virtue of the Vitali covering lemma \cite{FG} we give an estimation of the Hausdorff dimension of $Proj_θ(K_1\times K_2)$. For some cases, we can calculate, by means of Mauldin and Urbanski' result \cite{MRD}, the exact Hausdorff dimension of $Proj_θ(K_1\times K_2)$. We also find some non-trivial examples such that for some angle $θ\in[0,π)$ and some $K_1, K_2\in \mathcal{A}$, $\dim_{H}(Proj_θ(K_1\times K_2))=\dim_{H}(K_1)+\dim_{H}(K_2)$.
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