图有向马尔可夫系统的标度函数
Scaling functions for graph directed Markov systems
摘要：
我们引入了与图定向马尔可夫系统相关的尺度函数，并证明它是对偶符号康托尔集的Hölder连续函数。在一定的自然分离性和正则性条件下，每个系统在欧几里得空间中都有唯一的Cantor极限集。证明了尺度函数是极限集之间$ C^{1+α} $共轭的完全不变量。最后，我们将标度函数与压力联系起来，并讨论极限集的维数论的几个应用。
英文摘要：
We introduce the scaling function associated to a graph directed Markov system, and show that it is a Hölder continuous function of the dual symbolic Cantor set. With some natural separation and regularity conditions, each such system has a unique Cantor limit set in Euclidean space. We prove that the scaling function is a complete invariant of $ C^{1+α} $ conjugacy between limit sets. We conclude by relating the scaling function to the pressure, and discussing several applications to the dimension theory of limit sets.
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