微分动力系统中的维数理论以及重分形分析

本项目主要致力于三个方面的研究。其一是对$C^1$微分同胚，估计具有控制分解的非双曲不变测度的下点维数和上点维数，以及它们与系统其它不变量之间的关系。进一步，在一定的条件下讨论具有控制分解的双曲不变测度的点维数的存在性，也即Eckmann-Ruelle猜测是否成立？其二是讨论$C^1$微分同胚的平均共形双曲集的Hausdorr维数是否等于它限制在稳定方向上的Hausdorff维数与不稳定方向上的Hausdorrf维数之和。其三是研究$C^1$共形排斥子以及平均共形排斥子上相应的多重分形分析。. 申请人致力于动力系统维数理论与遍历理论的研究，已经在这方面得到若干有趣的结果，这将为本项目的创新成果提供保证。

This proposal will be divided into three parts. First of all, given $C^1$ diffeomorphism, we will give the estimates of the lower and upper pointwise dimension for an invariant measure with dominated splitting, and the relations between the fractal dimension of the measure and other invariants of the system. Moreover, we will study the existence of the pointwise dimension for a hyperbolic invariant probability measure with dominated splitting under some assumptions, which is still an open problem in dimension theory of dynamical systems and is often referred to as the Eckmann-Ruelle conjecture. Secondly, we will investigate the Hausdorff dimension of the average conformal hyperbolic set of $C^1$ diffeomorphism. Thirdly, we will effect a complete multifractal analysis of the non-linear dimension spectra for conformal repellers and average conformal repellers..The applicant committed to dimension theory of dynamic system and ergodic theory. And we have gotten some interesting results in this field, which ensure the proposal in progress and quality.