拓扑压和维数估计

Topological pressure and dimension estimation

本文综述了动力系统中维数理论的一些新进展,特别是对排斥子和双曲集的维数以及不变测度维数方面的介绍.在共形情形下,人们已经很好地理解了维数理论中的重要问题.在非共形情形下,虽然取得了许多有趣和非平凡的结果,但仍然缺乏一个一般的令人满意的方法.事实上,我们只对某些特殊的非共形排斥子的维数的理解比较清楚(如广义的Sierpin′ski毛毯和平均共形排斥子).

In this paper, we give a survey of recent results in the dimension theory of dynamical systems, with emphasis on the dimension of repellers and hyperbolic sets and the dimension of invariant measures. In the case of conformal dynamics, the theory is completely well understood. However, it still lacks today a satisfactory general approach for the non-conformal case, although a number of interesting and nontrivial developments have been obtained. Indeed, we only well understand some particular non-conformal repellers, e.g., generalized Sierpinski carpets and average conformal repellers.