We are interested in connections between the theory of fractal sets obtained as attractors of iterated function systems and process calculi. To this end, we reinterpret Milner's expressions for processes as contraction operators on a complete metric space. When the space is, for example, the plane, the denotations of fixed point terms correspond to familiar fractal sets. We give a sound and complete axiomatization of fractal equivalence, the congruence on terms consisting of pairs that construct identical self-similar sets in all interpretations. We further make connections to labelled Markov chains and to invariant measures. In all of this work, we use important results from process calculi. For example, we use Rabinovich's completeness theorem for trace equivalence in our own completeness theorem. In addition to our results, we also raise several questions related to both fractals and process calculi.

翻译：本文关注迭代函数系统吸引子所得到的分形集理论与进程演算之间的联系。为此，我们将米尔纳对进程的表达式重新解释为完备度量空间上的收缩算子。当空间为平面时，不动点项的含义对应于常见的分形集。我们给出了分形等价关系的完备公理化系统，该等价关系是项上的同余关系，包含在所有解释中构造相同自相似集的项对。此外，我们建立了与标号马尔可夫链及不变测度之间的联系。在整个工作中，我们利用了进程演算的重要结果。例如，在证明完备性定理时，我们应用了拉宾诺维奇关于迹等价的完备性定理。除所获成果外，我们还提出了若干与分形及进程演算相关的问题。