负β变换下的丢番图逼近问题研究

Diophantine approximations of dynamical systems arising from the number theory are not only related to metric and dimensional theory of expansions of numbers, but also hot topics of dynamical systems and fractals. We study the Diophantine approximations of the negative beta transformation. Different from the classical beta transformation from number theory, the negative beta transformation may not be topological transitive and the support of its invariant measure may not be the whole unit interval. Specifically, we study the following problems..1. We study the fine properties of the measure-theoretical and topological dynamical systems for the negative beta transformation. More precisely, we study the φ-mixing property of the invariant measure and the specification property of the negative beta transformation..2. We study Diophantine approximations of the negative beta transformation. It includes the distribution of orbits under the negative beta transformation, the recurrent problems of critical points, the shrinking targets and moving targets problems, and related fractals.

数论中动力系统的丢番图逼近问题,不仅与数的展式的度量和维数理论密切相关,也是分形几何和动力系统研究的热点问题之一。本项目研究负β变换下的丢番图逼近问题。与经典β变换不同,负β变换往往没有拓扑传递性,其不变测度的支撑也不一定是整个单位区间。我们的研究内容主要为:.一、负β变换测度和拓扑性质的精细刻画，即不变测度的φ-混合性和负β变换的specification性质及相关分形集。.二、负β变换下的丢番图逼近问题，主要包括负β变换下轨道的分布；负β变换下临界点的常返问题；以及负β变换下的收缩靶、移动靶和相关分形集。

数论中动力系统的丢番图逼近问题,不仅与数的展式的度量和维数理论密切相关,也是分形几何和动力系统研究的热点问题之一。本项目研究了负β变换下的丢番图逼近问题，同时也研究了一些其他数的展式字符分布的度量性质。与经典β变换不同,负β变换往往没有拓扑传递性,其不变测度的支撑也不一定是整个单位区间。我们的研究内容主要为:.一、负β变换测度和拓扑性质的精细刻画，即不变测度的φ-混合性和负β变换的specification性质及相关分形集。.二、负β变换下的丢番图逼近问题，主要包括负β变换下轨道的分布，负β变换下临界点的常返问题以及负β变换下的相关分形集。.同时，我们也研究了各类连分数： 经典连分数、Engel连分数展式、p-adic连分数、以及形式级数域上连分数展式字符分布的度量性质。