一维实或复动力系统中若干问题研究

Dynamical system is one of the important research directions in fundamental mathematics, and One-dimensional dynamics is one of the important research branch in dynamical system. The main objectives in one-dimensional dynamics are iterations of rational functions on Riemann sphere, or continuous functions on interval or circumference. The aims of this project are non-uniformly hyperbolic conditions, thermodynamical formalism and fractal properties of Julia sets and their invariant subset in one-dimensional real or complex dynamics. First, we will continue to study the relationship among various non-uniformly hyperbolic conditions and their invariant properties. Second, we will also prove some equalities of fractal dimensions of Julia sets for interval maps. Finally, we will pay more attention to the thermodynamical formalism in one-dimensional dynamics. Through these studies, we hope that we can understand the more properties of one-dimensional dynamics.

动力系统是基础数学的一个重要研究方向，一维动力系统又是动力系统中的一个重要研究分支，主要研究黎曼球面上的有理函数，或区间、圆周上的连续函数的迭代。本课题旨在研究一维实或复动力系统中的非一致双曲性条件，热力动力学性质以及Julia集及其不变子集的分形性质。首先，我们将延续申请人的青年基金项目，继续研究一维动力系统中各种非一致双曲性条件之间的关系和它们的不变形，同时也将证明区间映射中Julia集的分形维数之间的关系。其次，我们将重点考察一维动力系统中的热力动力学性质。希望通过上述研究，能够对一维动力系统有更新、更深刻的认识；能够更好地理解一维动力系统中的热力动力学性质。

本项目按计划书开展研究，较好地完成了预期的目标，得到了如下结果：我们研究了非一致双曲性条件，证明了Collect-Eckmann条件附件零界点的慢回复性是拓扑共轭不变的；我们研究了有理函数的Poincare级数，证明了对于一大类满足较弱双曲性条件的有理函数动力系统的 Poincare 临界指数和 Julia 集的双曲维数都是相等的；考察了区间映射的热力学动力性质，我们用平衡态测度的测度熵和李雅普指数给出了势函数是双曲的等价刻画。