Abstract
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 self similar sets by Kamo and Kawamura, and computability of the Takagi function.
This work has been supported in part by the Scientific Grant of Japan No. 12640120.
Acknowledgments
The author thanks to the participants of CCA 2000 for their helpful suggestions and comments.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
C. Bandt: Self-Similar Sets 3. Constructions with Sofic systems, Monatshefte für Mathematik, 108 (1989) 89–102.
A. Grzegorczyk: On the definition of computable real functions, Fund. Math. 44 (1957) 61–71.
M. Hata: On the structure of self-similar sets, Japan J. appl. Math., 2 (1985) 381–414.
J. E. Hutchinson: Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981) 713–747.
H. Kamo, K. Kawamura: Computability of self-similar sets, Math. Log. Quart. 45 (1999) 23–30.
T. Mori, Y. Tsujii, M. Yasugi: Computability structures on metric spaces, in: D. S. Bridges et al (Eds.), Combinatorics, Complexity and Logic, Proc. DMTCS’ 96, Springer, Berlin, (1996) 351–362.
M. B. Pour-El, J. I. Richards: Computability in Analysis and Physics, Springer-Verlag, Berlin, Heidelberg, 1989.
H. G. Rice: Recursive real numbers, Proc. Am. Math. Soc., 5 (1954) 784–791.
T. Takagi:A simple example of the continuous function without derivative, Proc. of the Physico-Mathematical Society of Japan, II-1 (1903) 176–177. Re-recorded in: The Collected Papers of Teiji Takagi, Iwanami Shoten Publ., Tokyo, (1973) 5-6.
K. Weihrauch: Computability, Springer-Verlag, Berlin, Heidelberg, 1987.
M. Yasugi, T. Mori, Y. Tsujii: Effective properties of sets and functions in metric spaces with computability structure, Theoret. Comput. Sci. 219 (1999) 467–486.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kamo, H. (2001). Effective Contraction Theorem and Its Application. In: Blanck, J., Brattka, V., Hertling, P. (eds) Computability and Complexity in Analysis. CCA 2000. Lecture Notes in Computer Science, vol 2064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45335-0_7
Download citation
DOI: https://doi.org/10.1007/3-540-45335-0_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42197-9
Online ISBN: 978-3-540-45335-2
eBook Packages: Springer Book Archive