关于共形迭代函数系极限集的几何结构
On the geometric structure of the limit set of conformal iterated function systems
摘要：
考虑$\mathbb{R}^d$上的无穷共形迭代函数系。我们研究了这类系统极限集的几何结构。设该极限集与某个具有正Hausdorff $t$ -维测度的$l$ -维$C^1$ -子流形相交，其中$0<l<d$和$t$是该极限集的豪斯多夫维数.证明了当$d$超过$2$时，极限集的闭包属于某个$l$维仿射子空间或几何球面;当$d$等于$2$时，极限集的闭包属于解析曲线.
英文摘要：
We consider infinite conformal iterated function systems on $\mathbb{R}^d$. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some $l$-dimensional $C^1$-submanifold with positive Hausdorff $t$-dimensional measure, where $0<l<d$ and $t$ is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs to some $l$-dimensional affine subspace or geometric sphere whenever $d$ exceeds $2$ and analytic curve if $d$ equals $2$.
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