Iterations of rational functions: which hyperbolic components contain polynomials?

• Autorzy: Feliks Przytycki
• Języki publikacji: EN

Abstrakty

EN

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 exotic} examples 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

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1996
• Tom: 149
• Numer: 2
• Strony: 95-118

Daty

wydano

1996

otrzymano

1994-04-07

poprawiono

1995-07-14

Twórcy

autor

Feliks Przytycki

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warszawa, Poland

Bibliografia

• [AB] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. 72 (1960), 385-404.
• [Ba] K. Barański, PhD thesis, in preparation.
• [B] B. Bojarski, Generalized solutions of systems of differential equations of first order and elliptic type with discontinuous coefficients, Mat. Sb. 43 (85) (1957), 451-503 (in Russian).
• [Boy] M. Boyle, a letter.
• [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.
• [D] A. Douady, Chirurgie sur les applications holomorphes, in: Proc. ICM Berkeley 1986, 724-738.
• [DH1] A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. 18 (1985), 287-243.
• [DH2] A. Douady and J. Hubbard, Étude dynamique des polynômes complexes, Publ. Math. Orsay 2 (1984), 4 (1985).
• [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.
• [M] P. Makienko, Pinching and plumbing deformations of quadratic rational maps, preprint, Internat. Centre Theoret. Phys., Miramare-Trieste, 1993.
• [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.