Iterations of rational functions: which hyperbolic components contain polynomials?

Feliks Przytycki

Fundamenta Mathematicae (1996)

  • Volume: 149, Issue: 2, page 95-118
  • ISSN: 0016-2736

Abstract

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Let H d be the set of all rational maps of degree d ≥ 2 on the Riemann sphere, expanding on their Julia set. We prove that if f ∈ H d and all, or all but one, critical points (or values) are in the basin of immediate attraction to an attracting fixed point then there exists a polynomial in the component H(f) of H d containing f. If all critical points are in the basin of immediate attraction to an attracting fixed point or a parabolic fixed point then f restricted to the Julia set is conjugate to the shift on the one-sided shift space of d symbols. We give exoticexamples of maps of an arbitrary degree d with a non-simply connected completely invariant basin of attraction and arbitrary number k ≥ 2 of critical points in the basin. For such a map f ∈ H d with k

How to cite

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Przytycki, Feliks. "Iterations of rational functions: which hyperbolic components contain polynomials?." Fundamenta Mathematicae 149.2 (1996): 95-118. <http://eudml.org/doc/212117>.

@article{Przytycki1996,
abstract = {Let $H^d$ be the set of all rational maps of degree d ≥ 2 on the Riemann sphere, expanding on their Julia set. We prove that if $f ∈ H^d$ and all, or all but one, critical points (or values) are in the basin of immediate attraction to an attracting fixed point then there exists a polynomial in the component H(f) of $H^d$ containing f. If all critical points are in the basin of immediate attraction to an attracting fixed point or a parabolic fixed point then f restricted to the Julia set is conjugate to the shift on the one-sided shift space of d symbols. We give exoticexamples of maps of an arbitrary degree d with a non-simply connected completely invariant basin of attraction and arbitrary number k ≥ 2 of critical points in the basin. For such a map $f ∈ H^d$ with k },
author = {Przytycki, Feliks},
journal = {Fundamenta Mathematicae},
keywords = {rational functions; iterations},
language = {eng},
number = {2},
pages = {95-118},
title = {Iterations of rational functions: which hyperbolic components contain polynomials?},
url = {http://eudml.org/doc/212117},
volume = {149},
year = {1996},
}

TY - JOUR
AU - Przytycki, Feliks
TI - Iterations of rational functions: which hyperbolic components contain polynomials?
JO - Fundamenta Mathematicae
PY - 1996
VL - 149
IS - 2
SP - 95
EP - 118
AB - Let $H^d$ be the set of all rational maps of degree d ≥ 2 on the Riemann sphere, expanding on their Julia set. We prove that if $f ∈ H^d$ and all, or all but one, critical points (or values) are in the basin of immediate attraction to an attracting fixed point then there exists a polynomial in the component H(f) of $H^d$ containing f. If all critical points are in the basin of immediate attraction to an attracting fixed point or a parabolic fixed point then f restricted to the Julia set is conjugate to the shift on the one-sided shift space of d symbols. We give exoticexamples of maps of an arbitrary degree d with a non-simply connected completely invariant basin of attraction and arbitrary number k ≥ 2 of critical points in the basin. For such a map $f ∈ H^d$ with k
LA - eng
KW - rational functions; iterations
UR - http://eudml.org/doc/212117
ER -

References

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  4. [Boy] M. Boyle, a letter. 
  5. [CGS] J. 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
  6. [D] A. Douady, Chirurgie sur les applications holomorphes, in: Proc. ICM Berkeley 1986, 724-738. 
  7. [DH1] A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. 18 (1985), 287-243. Zbl0587.30028
  8. [DH2] A. Douady and J. Hubbard, Étude dynamique des polynômes complexes, Publ. Math. Orsay 2 (1984), 4 (1985). 
  9. [GK] L. Goldberg and L. Keen, The mapping class group of a generic quadratic rational map and automorphisms of the 2-shift, Invent. Math. 101 (1990), 335-372. Zbl0715.58018
  10. [M] P. Makienko, Pinching and plumbing deformations of quadratic rational maps, preprint, Internat. Centre Theoret. Phys., Miramare-Trieste, 1993. 
  11. [P] F. Przytycki, Remarks on simple-connectedness of basins of sinks for iterations of rational maps, in: Banach Center Publ. 23, PWN, 1989, 229-235. Zbl0703.58033

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