Non-computable Julia sets
Journal of the American Mathematical Society, Tome 19 (2006) no. 3, pp. 551-578

Voir la notice de l'article provenant de la source American Mathematical Society

We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
DOI : 10.1090/S0894-0347-05-00516-3

Braverman, M. 1 ; Yampolsky, M. 2

1 Department of Computer Science, University of Toronto, Toronto, ON M5S 3G4, Canada
2 Department of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
@article{10_1090_S0894_0347_05_00516_3,
     author = {Braverman, M. and Yampolsky, M.},
     title = {Non-computable {Julia} sets},
     journal = {Journal of the American Mathematical Society},
     pages = {551--578},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {2006},
     doi = {10.1090/S0894-0347-05-00516-3},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-05-00516-3/}
}
TY  - JOUR
AU  - Braverman, M.
AU  - Yampolsky, M.
TI  - Non-computable Julia sets
JO  - Journal of the American Mathematical Society
PY  - 2006
SP  - 551
EP  - 578
VL  - 19
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-05-00516-3/
DO  - 10.1090/S0894-0347-05-00516-3
ID  - 10_1090_S0894_0347_05_00516_3
ER  - 
%0 Journal Article
%A Braverman, M.
%A Yampolsky, M.
%T Non-computable Julia sets
%J Journal of the American Mathematical Society
%D 2006
%P 551-578
%V 19
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-05-00516-3/
%R 10.1090/S0894-0347-05-00516-3
%F 10_1090_S0894_0347_05_00516_3
Braverman, M.; Yampolsky, M. Non-computable Julia sets. Journal of the American Mathematical Society, Tome 19 (2006) no. 3, pp. 551-578. doi: 10.1090/S0894-0347-05-00516-3

[1] Blum, Lenore, Cucker, Felipe, Shub, Michael, Smale, Steve Complexity and real computation 1998

[2] Avila, Artur, Buff, Xavier, Chã©Ritat, Arnaud Siegel disks with smooth boundaries Acta Math. 2004 1 30

[3] Bishop, Errett, Bridges, Douglas Constructive analysis 1985

[4] Brjuno, A. D. Analytic form of differential equations. I, II Trudy Moskov. Mat. Obšč. 1971

[5] Douady, Adrien Does a Julia set depend continuously on the polynomial? 1994 91 138

[6] Douady, A., Hubbard, J. H. Étude dynamique des polynômes complexes. Partie I 1984 75

[7] Douady, Adrien, Hubbard, John Hamal On the dynamics of polynomial-like mappings Ann. Sci. École Norm. Sup. (4) 1985 287 343

[8] Grzegorczyk, A. Computable functionals Fund. Math. 1955 168 202

[9] Ko, Ker-I Complexity theory of real functions 1991

[10] Ko, K. Polynomial-time computability in analysis 1998 1271 1317

[11] Lacombe, Daniel Extension de la notion de fonction récursive aux fonctions d’une ou plusieurs variables réelles. I C. R. Acad. Sci. Paris 1955 2478 2480

[12] Mazur, S. Computable analysis Rozprawy Mat. 1963 110

[13] Marmi, S., Moussa, P., Yoccoz, J.-C. The Brjuno functions and their regularity properties Comm. Math. Phys. 1997 265 293

[14] Matiyasevich, Yu. V. Desyataya problema Gil′berta 1993 224

[15] Mcmullen, Curtis T. Complex dynamics and renormalization 1994

[16] Milnor, John Dynamics in one complex variable 1999

[17] Petersen, C. L., Zakeri, S. On the Julia set of a typical quadratic polynomial with a Siegel disk Ann. of Math. (2) 2004 1 52

[18] Pommerenke, Ch. Boundary behaviour of conformal maps 1992

[19] Rettinger, Robert, Weihrauch, Klaus The computational complexity of some Julia sets 2003 177 185

[20] Rohde, Steffen, Zinsmeister, Michel Variation of the conformal radius J. Anal. Math. 2004 105 115

[21] Siegel, Carl Ludwig Iteration of analytic functions Ann. of Math. (2) 1942 607 612

[22] Weihrauch, Klaus Computable analysis 2000

[23] Yoccoz, Jean-Christophe Théorème de Siegel, nombres de Bruno et polynômes quadratiques Astérisque 1995 3 88

Cité par Sources :