# Iterations of rational functions: which hyperbolic components contain polynomials?

Fundamenta Mathematicae (1996)

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

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topPrzytycki, 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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