Computability
Effective properties of sets and functions in metric spaces with computability structure
Theoretical Computer Science - Special issue on computability and complexity in analysis
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A contraction on a complete metric space has a unique fixed point. This fact is called the contraction theorem and of wide application. In this paper, we present an effective version of the contraction theorem. We show that if the contraction is a computable function on an effectively locally compact metric space, then the fixed point is a computable point on the space. Many facts on computability can be proved by using the effective contraction theorem. We give, in this paper, three examples, the effective implicit function theorem, the result on computability of selfsimilar sets by Kamo and Kawamura, and computability of the Takagi function.