返回时间的全局统计：返回时间维与广义测度维
The global statistics of return times: return time dimensions versus generalized measure dimensions
摘要：
研究了动力系统中不变测度的广义维数与返回时间幂的全局平均标度所定义的相似量之间的关系。由于卡茨定理的启发式使用，后者已被用来代替前者在数值和实验研究;为了标记这种区别，我们称之为返回时间维。我们推导出一整套将测度和返回时间维度联系起来的不等式，并借助冯·诺伊曼-角谷和加斯帕德-王的两个映射来评论它们的最优性。我们推测在一个典型的系统中的返回时间维度的行为。我们只假设所研究的动力系统是遍历的。
英文摘要：
We investigate the relations holding among generalized dimensions of invariant measures in dynamical systems and similar quantities defined by the scaling of global averages of powers of return times. Because of a heuristic use of Kac theorem, these latter have been used in place of the former in numerical and experimental investigations; to mark this distinction, we call them return time dimensions. We derive a full set of inequalities linking measure and return time dimensions and we comment on their optimality with the aid of two maps due to von Neumann -- Kakutani and to Gaspard -- Wang. We conjecture the behavior of return time dimensions in a typical system. We only assume ergodicity of the dynamical system under investigation.
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