From Newton’s method to exotic basins Part I: The parameter space

Krzysztof Barański

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 3, page 249-288
  • ISSN: 0016-2736

Abstract

top
This is the first part of the work studying the family 𝔉 of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of 𝔉 and give a detailed study of the subfamily ℱ 2 consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in ℱ 2 from Newton maps to maps with so-called exotic basins.

How to cite

top

Barański, Krzysztof. "From Newton’s method to exotic basins Part I: The parameter space." Fundamenta Mathematicae 158.3 (1998): 249-288. <http://eudml.org/doc/212315>.

@article{Barański1998,
abstract = {This is the first part of the work studying the family $\mathfrak \{F\}$ of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of $\mathfrak \{F\}$ and give a detailed study of the subfamily $ℱ_2$ consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in $ℱ_2$ from Newton maps to maps with so-called exotic basins.},
author = {Barański, Krzysztof},
journal = {Fundamenta Mathematicae},
keywords = {superattracting fixed points; moduli space; exotic basins; Julia sets; bifurcations; rational maps; Newton maps; parabolic bifurcation; Mandelbrot-like sets},
language = {eng},
number = {3},
pages = {249-288},
title = {From Newton’s method to exotic basins Part I: The parameter space},
url = {http://eudml.org/doc/212315},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Barański, Krzysztof
TI - From Newton’s method to exotic basins Part I: The parameter space
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 3
SP - 249
EP - 288
AB - This is the first part of the work studying the family $\mathfrak {F}$ of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of $\mathfrak {F}$ and give a detailed study of the subfamily $ℱ_2$ consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in $ℱ_2$ from Newton maps to maps with so-called exotic basins.
LA - eng
KW - superattracting fixed points; moduli space; exotic basins; Julia sets; bifurcations; rational maps; Newton maps; parabolic bifurcation; Mandelbrot-like sets
UR - http://eudml.org/doc/212315
ER -

References

top
  1. [Ba] K. Barański, Connectedness of the basin of attraction for rational maps, Proc. Amer. Math. Soc. (6) 126 (1998), 1857-1866. Zbl0898.58043
  2. [BH] B. Branner and J. Hubbard, The iteration of cubic polynomials, II: Patterns and parapatterns, Acta Math. 169 (1992), 229-325. Zbl0812.30008
  3. [CGS] J. H. Curry, L. Garnett and D. Sullivan, On the iteration of a rational function: computer experiments with Newton's method, Comm. Math. Phys. 91 (1983), 267-277. Zbl0524.65032
  4. [DH1] A. Douady et J. H. Hubbard, Etude dynamique des polynômes complexes, I et II, avec la collaboration de P. Lavours, Tan Lei et P. Sentenac, Publication d'Orsay 84-02, 85-04, 1984-1985. 
  5. [DH2] A. Douady et J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), 287-343. Zbl0587.30028
  6. [HP] F. von Haeseler and H.-O. Peitgen, Newton's method and complex dynamical systems, Acta Appl. Math. 13 (1988), 3-58. Zbl0671.30023
  7. [He] J. Head, The combinatorics of Newton's method for cubic polynomials, Ph.D. thesis, Cornell University, Ithaca, 1987. 
  8. [MSS] R. Ma né, P. Sad and D. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), 193-217. 
  9. [Mi1] J. Milnor, Dynamics in one complex variable: introductory lectures, preprint, SUNY at Stony Brook, IMS # 1990/5. 
  10. [Mi2] J. Milnor, Geometry and dynamics of quadratic rational maps, Experiment. Math. 2 (1993), 37-83. Zbl0922.58062
  11. [P1] F. Przytycki, Iterations of rational functions: which hyperbolic components contain polynomials?, Fund. Math. 149 (1996), 95-118. Zbl0852.58052
  12. [P2] F. Przytycki, Remarks on simple-connectedness of basins of sinks for iterations of rational maps, in: Banach Center Publ. 23, PWN, 1989, 229-235. 
  13. [Re1] M. Rees, A partial description of parameter space of rational maps of degree two, I: Acta Math. 168 (1992), 11-87; II: Proc. London Math. Soc. (3) 70 (1995), 644-690. Zbl0774.58035
  14. [Re2] M. Rees, Components of degree two hyperbolic rational maps, Invent. Math. 100 (1990), 357-382. Zbl0712.30022
  15. [Ro] P. Roesch, Topologie locale des méthodes de Newton cubiques, Ph.D. thesis, École Norm. Sup. de Lyon, 1997. 
  16. [Se] G. Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), 39-72. Zbl0427.55006
  17. [Sh] M. Shishikura, The connectivity of the Julia set of rational maps and fixed points, preprint, Inst. Hautes Études Sci., Bures-sur-Yvette, 1990. 
  18. [Ta] Tan, Branched coverings and cubic Newton maps, Fund. Math. 154 (1997), 207-260. Zbl0903.58029

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.