重分形分解中的不正则集

The irregular set arises naturally in the context of multifractal analysis.It can recover some information of the system.Therefore it is of great significance to investigate the irregular sets in the dynamical system and fractal geometry. there are lots of excellent results on the dimensions of the irregular sets.It has been proposed by Bonatti, Diaz and Viana in their book 《Dynamics Beyond Uniform Hyperbolicity》that in order to study the statistical complexity of the system one might consider the higher order average of the local invariant quantities associated with the system. Following this framework, we have to face the higher order irregular sets. Baire categry is an important tool to describe the size of set in the topological sense.In this project we plan to study the irregular sets (as well as the higher order irregular sets) from the topological point of view and the higher order irregular sets from the point of dimension theory (Hausdorff dimension, topological entropy). Moreover, we will apply our results on repeller and hyperbolic set. We study the irregular sets from a new angle. Moreover, the study makes us understand better the complexity of the system and the (topological) structure of the irregular sets.On the other hand, our study will supply substantial materials to the theory on fractal and dynamical system and makes it more perfect.

不正则集是重分形分析中自然出现的一类集合，它可以还原系统的某些信息。因此，不正则集的研究在动力系统和分形几何中具有重要意义。已有大量出色的工作从维数角度研究不正则集。Bonatti,Diaz和Viana在其专著《Dynamics Beyond Uniform Hyperbolicity》中提出：为了进一步了解系统的某些统计复杂性，需要考虑与系统相关的局部量的高阶平均。在此研究过程中自然会产生高阶不正则集。纲性是从拓扑角度来刻画集合"大小"的一个重要工具。本项目计划对不正则集（包含高阶不正则集）的纲性和高阶不正则集的满维性质（Hausdorff 维数、拓扑熵）进行研究，并将所得结果应用到斥子和双曲集上。本项目为不正则集的研究提供了一个新的视角，使我们不仅可以更好地描述系统的复杂性，而且对不正则集的（拓扑）结构有更深入的了解。同时，使重分形分析及动力系统的理论得到本质地补充和完善。

不正则集是重分形分析中自然出现的一类集合，它可以还原系统的某些信息。 因此，不正则集的研究在动力系统和分形几何中具有重要意义。本项目从维数和纲的角度对分形几何、动力系统和概率中出现的不正则集进行研究. 另外，为了更精细地研究不正则集的结构，我们引进了极大Birkhoff遍历平均振幅这一概念。我们得到了一序列结果和应用。本项目得研究为不正则集的研究提供了一个新的视角，使我们不仅可以更好地描述系统的复杂性，而且对不正则集的（拓扑）结构有更深入的了解。同时，使重分形分析及动力系统的理论得到本质地补充和完善。