Scaling functions for graph directed Markov systems
Abstract
We introduce the scaling function associated to a graph directed Markov system, and show that it is a H\"older 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 C1+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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