连分数和beta-展式中存活集的维数研究

This proposal concerns the problems related to the survivor set in continued fraction and beta-expansions. The exact object of this research is the survivor set induced by the dynamical system with a hole. The survivor set consists of the points whose orbit never enter the hole. Such set can be regarded as a Cantor-like set in fractal geometry (maybe infinitely many overlaps for the removing parts), and also can be regarded as badly approximate set in Diophantine approximation. We study the continuity of the Hausdorff dimension of the survivor set about the size and position of the hole. We hope to obtain such continuity and furthermore the Hausdorff dimension of the survivor set. Meanwhile we study the relationship between the Hausdorff dimension of the survivor set and escape rate, and the formula linking them and Lyapunov exponent. We will extend the above results in continued fraction and beta-dynamical systems to general piecewise monotonic expanding dynamical systems on the unit interval. We will use the tools of the measure and dimension theory of fractal sets, the symbolic representations of invariant sets and variational principle etc. This project concerns fractal geometry, dynamical system and number theory, and will make these fields closer and closer.

本项目研究连分数和beta-展式中存活集的相关问题，在单位区间中加入一个洞，考察轨道一直不进入该洞的所有点组成的集合，存活集是一种类Cantor集，但挖去的部分可能重叠无穷多次，也是动力丢番图逼近中的不可很好逼近集。我们将研究存活集的Hausdorff维数对洞的大小和位置的敏感依赖性，讨论该维数关于洞的半径和中心的连续性，希望得到该维数值；同时也探讨该维数与逃逸率之间的内在联系，希望建立起这两者与Lyapunov指数的关系式。进一步地将连分数和beta-变换动力系统中的这两类问题推广到单位区间上分段单调的扩张动力系统。我们将综合运用分形集的测度和维数理论、不变集的符号表示、变分原理等工具来解决这些问题，该项目涉及分形几何、动力系统、度量数论等研究领域，将促进这些领域间的相互联系。

我们对连分数、beta-展式及beta-连分数中的存活集相关问题进行研究，探讨其相应动力系统轨道的丢番图逼近性质。得到了连分数的distal集和渐近集的测度和Hausdorff维数，对连分数动力系统，证明存在平均Li-Yorke攀援集(scrambled set)是满Hausdorff维数的，这可以得到该系统的平均Li-Yorke混沌性。证明了beta-变换动力系统中的不可很好逼近集是满Hausdorff维数的，并证明该系统的distal集在Schmidt博弈意义下总是赢集。进一步地对beta-连分数动力系统轨道渐近行为有关的丢番图性质进行了研究。这些结果发表在Fractals、Monatsh. Math.、Dyn. Syst.等国际SCI期刊上，共6篇。研究过程中综合运用了分形集的测度和维数理论、不变集的符号表示、变分原理等工具，促进分形几何、动力系统、度量数论等研究领域的相互联系。