On the computability of fractal dimensions and Hausdorff measure

Annals of Pure and Applied Logic

Volumn 93, Issue 1-3, 1998, Pages 195-216

  Ko, Ker I ^a  

^a Stony Brook University   (United States)

Author keywords

Hausdorff dimension; Julia sets; Polynomial time computable real functions; Recursive real numbers; Recursively approximable sets

Indexed keywords

EID: 0032574619     PISSN: 01680072     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-0072(97)00060-2     Document Type: Article

References (16)

1. [1] O. Aberth, Computable Analysis, McGraw-Hill, New York, 1980.
2. [2] L. Blum, M. Shub, S. Smale, On a theory of computation and complexity over the real numbers; NP completeness, recursive functions and universal machines, Bull. Amer. Math. Soc. 21 (1989) 1-46.
3. [3] J.-Y. Cai, J. Hartmanis, On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line, J. Comput. System Sci. 49 (1994) 605-619.
4. [4] D. Cenzer, Effective real dynamics, in: J.N. Crossley, J.B. Remmel, R.A. Shore, M.E. Sweedler (Eds.), Logical Methods, Birkhäuser, Boston, 1993, pp. 162-177.
5. [5] A. Chou, K. Ko, Computational complexity of two-dimensional regions, SIAM J. Comput. 24 (1995) 923-947.
6. [6] K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, New York, 1991.
7. [7] R. Goodstein, Recursive Analysis, North-Holland, Amsterdam, 1961.
8. [8] K. Ko, Complexity Theory of Real Functions, Birkhäuser, Boston, 1991.
9. [9] K. Ko, A polynomial-time computable curve whose interior has a nonrecursive measure, Theoret. Comput. Sci. 145 (1995) 241-270.
10. [10] K. Ko, K. Weihrauch, On the measure of two-dimensional regions with polynomial-time computable boundaries, in: Proc. 11th IEEE Conf. on Computational Complexity, 1996, pp. 150-159.
11. [11] D. Lacombe, Les ensembles récursivement ouverts on fermès et leurs applications à l'analyse récursive, Comptes Rendus 244 (1957) 838-840.
12. [12] B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman, New York, 1983.
13. [13] W. Merzenich, L. Staiger, Fractals, dimension, and formal languages, RAIRO Inform. Théor. 28 (1994) 361-386.
14. [14] M. Pour-El, I. Richards, Computability in Analysis and Physics, Springer, Berlin, 1989.
15. [15] N. Šanin, Constructive Real Numbers and Function Spaces, Translations of Mathematical Monographs, vol. 21 (transl. by E. Mendelson), American Mathematical Society, Providence, RI, 1968.
16. [16] K. Weihrauch, Computability, Springer, Berlin, 1987.