分形集在相似扩张映射下的性质

This project plans to study the properties of the sequence of image sets of a fractal set under a similar expanding mapping. The research is divided into two parts. .The first part is to study the limit behavior of image sets on the circle. Suppose that p,q are multiplicatively independent, Furstenberg conjectured that, for any p-invariant set A, T_q^n(A) tends to the circle under the Hausdorff metric, where T_q(x)=qx mod1. We plan to study the problems closely related to this conjecture from four different perspectives..The second part is to study the intersection of these image sets in Euclidean space. We focus on studying image sets of self-similar sets. For this, introducing the concepts of similar sequence and similarity ratio of it, we translate the problem to: if a self-similar set with the SSC contains a similar sequence, is there a logarithmic rational correlation between the similarity ratio of the similar sequence and the ratios of the self-similar set? Since this problem is closely related to the affine embedding problem of self-similar sets, we plan to study it by using techniques which appear in the study of the latter.

本项目计划研究分形集在相似扩张映射不断作用下所得到的像集序列的性质。研究分为两个部分。.第一部分是在圆周上研究像集序列的极限行为。假设p,q乘法独立，Furstenberg猜测：任何乘p不变集在不断乘q作用下，依Hausdorff度量趋于整个圆周。我们计划从4个不同的角度研究与这个猜测密切相关的问题。.第二部分是在欧式空间上研究这些像集的交。我们着重于研究自相似集的像集，为此引入相似序列及其相似比的概念，将问题转化为：若强分离自相似集包含某个相似序列，该相似序列的相似比与自相似集的压缩比之间是否存在对数有理相关性？因这个问题与自相似集仿射嵌入问题密切相关，我们计划用研究后者的技巧来研究它。

本项目的研究目标是讨论某些分形集在相似扩张映射下像集的极限行为，以及与这些像集的交集相关的结构问题。为此，我们研究了以下的具体问题。.与像集极限行为相关的：数字限制集一致分布问题；某些扩张映射诱导的轨道的渐进行为及相应的Diophantine逼近问题；某些分形集的Assouad维数与相关维谱问题。与交集相关的结构问题：数字限制集等差子列的研究；自相似集Lipschitz等价的不变量；分形方块的连通性质；由二进扩张映射诱导的随机分形集与解析集的碰撞问题。.上述这些具体问题的研究结果加深了我们对这些分形集在相似扩张映射下的像集的理解，使得我们能够在相当程度上刻画这些像集的极限行为及其交集的结构。