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We introduce the scaling function associated to a graph directed Markovsystem, and show that it is a H\"{o}lder continuous function of the dualsymbolic Cantor set. With some natural separation and regularity conditions,each such system has a unique Cantor limit set in Euclidean space. We provethat the scaling function is a complete invariant of $ C^{1+\alpha} $ conjugacybetween limit sets. We conclude by relating the scaling function to thepressure, and discussing several applications to the dimension theory of limitsets.

Scaling functions for graph directed Markov systems