Hiroyasu Kamo

Computability of Self-Similar Sets

We investigate computability of a self-similar set on a Euclidean space. A nonempty compact subset of a Euclidean space is called a self-similar set if it equals to the union of the images of itself by some set of contractions. The main result in this paper is that if all of the contractions are computable, then the self-similar set is a recursive compact set. A further result on the case that the self-similar set forms a curve is also discussed.

Effective Dini's Theorem on Effectively Compact Metric Spaces

We show that if a computable sequence of real-valued functions on an effectively compact metric space converges pointwise monotonically to a computable function, then the sequence converges effectively uniformly to the function. This is an effectivized version of Dinis Theorem.

Effective Contraction Theorem and Its Application

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.