# zbMATH — the first resource for mathematics

Fixed points of polynomial maps. II: Fixed point portraits. (English) Zbl 0771.30028
[For a review of part I see the preceding review.]
Let $$f$$ be a monic polynomial whose filled-in Julia set $$K(f)$$ is connected and denote by $$\varphi$$ the conformal map from $$\widehat {\mathbb{C}}\setminus D$$ to $$\widehat {\mathbb{C}}\setminus K$$ which is asymptotic to the identity at $$\infty$$. It turns out that (1) $$\varphi(z^ d)=f(\varphi(z))$$. For $$t\in\mathbb{R}\setminus\mathbb{Z}$$ the external ray $$R_ t=\{\varphi(r\exp(2\pi it))$$, $$1<r<-\infty\}$$. If $$t$$ is rational $$R_ t$$ lands at a point $$\xi=\lim_{r\to 1}\varphi(r\exp(2\pi t))$$ in the Julia set $$J(f)$$. Every repelling or parabolic fixed point of $$f$$ is the landing point of a finite (non-empty) set of external rays. If no rational external rays land at a fixed point (for example because it is attracting), the point is called rationally invisible. By (1) the rays with $$t=j/(d-1)$$, $$0\leq j<d$$, are fixed under $$f$$. If at least one of these last rays lands at a fixed point, then it is said to have rotation number $$\rho=0$$ and its type $$T$$ is the set of such fixed rays which land at the point. Finally, if the set $$T$$ of rational rays which land at a fixed point is non-empty and does not contain fixed rays, then $$T$$ forms a degree $$d$$ rotation set in the sense of Part I and has some rational combinatorial rotation number $${p\over q}\neq 0$$ in $$\mathbb{Q}/\mathbb{Z}$$. The fixed point portrait of $$f$$ is the collection $$P=\{T_ 1,\dots,T_ k\}$$ of the types of its rationally visible fixed points. The authors obtain four necessary conditions for $$P$$:
P1. Each $$T_ j$$, $$1\leq j\leq k$$ ($$\leq d$$) is a degree $$d$$ rational rotation set with some well-defined rotation number $$\rho_ j$$.
P2. If $$i\neq j$$, $$T_ i\subset\;$$ a single connected component of $$(\mathbb{R}/\mathbb{Z})-T_ j$$.
P3. The union of $$T_ j$$ with rotation number zero is the set of fixed external rays.
P4. If $$T_ i$$, $$T_ j$$ have non-zero rotation numbers and $$i\neq j$$, then $$T_ i$$, $$T_ j$$ are separated in $$\mathbb{R}/\mathbb{Z}$$ by at least one $$T_ \ell$$ with rotation number zero.
The question arises whether these conditions are also sufficient and the main result of the paper is that this is indeed so if $$k=d$$: given $$d$$ non-vacuous rational types satisfying P1-4 there is a critically preperiodic polynomial of degree $$d$$ for which this is the fixed point portrait. (The general case has been solved subsequently by A. Poirier, Stony Brook I.M.S., Preprint 1991/20).
Appendices give extensions to cases where the Julia set is not connected, to examining the change in the fixed point portrait under change in the parameters of $$f$$ and to applying the results in the special case of degree two.
Reviewer: I.N.Baker (London)

##### MSC:
 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
##### Keywords:
external rays; Julia st; fixed point
Full Text:
##### References:
  P. ATELA , The Mandelbrot Set and \sigma -Automorphisms of Quotients of the Shift , preprint PAM # 20, Applied Math, U. of Colorado. · Zbl 0766.30019 · doi:10.2307/2154400  P. ATELA , Bifurcation of Dynamical Rays in Complex Polynomials of Degree two , preprint PAM # 52, Applied Math, U. of Colorado. · Zbl 0768.58034  B. BIELEFELD , Conformal Dynamics Problem List , Stony Brook Institute for Mathematical Sciences Preprint, 1990 / 1991 .  P. BLANCHARD , Complex Analytic Dynamics on the Riemann Sphere (Bull. Amer. Math. Soc., Vol. 11, 1984 , pp. 84-141). Article | MR 85h:58001 | Zbl 0558.58017 · Zbl 0558.58017 · doi:10.1090/S0273-0979-1984-15240-6 · minidml.mathdoc.fr  B. BRANNER , The Mandelbrot Set , in Chaos and Fractals, DEVANEY and KEEN Eds. (A.M.S., Proc. Symp. Applied Math., Vol. 39, 1989 ). MR 1010237  R. F. BROWN , The Lefschetz Fixed Point Theorem , Scott-Foresman 1971 . MR 44 #1023 | Zbl 0216.19601 · Zbl 0216.19601  B. BRANNER and A. DOUADY , Surgey on Complex Polynomials (to appear). · Zbl 0668.58026  B. BIELEFELD , Y. FISHER and J. H. HUBBARD , manuscript in preparation.  B. BRANNER and J. H. HUBBARD , The Iteration of Cubic Polynomials , Part I, Acta Math., 1989 ; Part II, to appear. · Zbl 0668.30008  A. DOUADY , Systèmes dynamiques holomorphes (Séminaire Bourbaki, 35e année 1982 / 1983 , No. 599). Numdam | Zbl 0532.30019 · Zbl 0532.30019 · numdam:SB_1982-1983__25__39_0 · eudml:110016  A. DOUADY , Julia Sets and the Mandelbrot Set , pp. 161-173 of The Beauty of Fractals, PEITGEN and RICHTER Eds., Springer, 1986 . · Zbl 0603.30030  R. DEVANEY , Introduction to Chaotic Dynamical Systems , Addison-Wesley, 1985 , 1989 . Zbl 0695.58002 · Zbl 0695.58002  J. DUGUNDJI and A. GRANAS , Fixed Point Theory , Polish Scientific Publishers, Warsaw, 1982 . MR 83j:54038 | Zbl 0483.47038 · Zbl 0483.47038  A. DOUADY and J. H. HUBBARD , Itération des polynômes quadratiques complexes (C. R. Acad. Sci. Paris, T. 294, Séries I, 1982 , pp. 123-126). MR 83m:58046 | Zbl 0483.30014 · Zbl 0483.30014  A. DOUADY and J. H. HUBBARD , Étude dynamique des polynômes complexes , Parts I, II, Publ. Math. Orsay, 1984 - 1985 . Article | Zbl 0552.30018 · Zbl 0552.30018 · minidml.mathdoc.fr  A. DOUADY and J. H. HUBBARD , A proof of Thurston’s Topological Characterization of Rational Functions , preprint, Mittag-Leffler 1984 . · Zbl 0806.30027  A. DOUADY and J. H. HUBBARD , On the Dynamics of Polynomial-like Mappings (Ann. Sci. Ec. Norm. Sup., Vol. 18, 1985 , pp. 287-343). Numdam | MR 87f:58083 | Zbl 0587.30028 · Zbl 0587.30028 · numdam:ASENS_1985_4_18_2_287_0 · eudml:82160  W. DE MELO , Lectures on One-Dimensional Dynamics (17^\circ Colóq. Brasil. Mat., I.M.P.A., 1989 ).  Y. FISHER , Thesis , Cornell Univ. 1989 .  A. GRANAS , The Leray-Schauder Index and the Fixed Point Theory for arbitrary ANR’s (Bull. Soc. Math. France, Vol. 100, 1972 , pp. 209-228). Numdam | MR 46 #8213 | Zbl 0236.55004 · Zbl 0236.55004 · numdam:BSMF_1972__100__209_0 · eudml:87184  S. HU and Y. JIANG , Towards Topological Classification of Critically Finite Polynomials (in preparation). · Zbl 0846.57002  B. JIANG , Lectures on Nielsen Fixed Point Theory (Contemp. Math., Vol. 14, A.M.S., 1983 ). MR 84f:55002 | Zbl 0512.55003 · Zbl 0512.55003 · doi:10.1090/conm/014  S. LEVY , Critically Finite Rational Maps (Thesis, Princeton University, 1985 ).  M. LYUBICH , The Dynamics of Rational Transforms : The Topological Picture (Russ. Math. Surv., Vol. 41, 4, 1986 , pp. 43-117). Zbl 0619.30033 · Zbl 0619.30033 · doi:10.1070/RM1986v041n04ABEH003376  J. MILNOR , Self-Similarity and Hairiness in the Mandelbrot set , pp. 211-257 of Computers in Geometry and Topology, TANGORA Ed., Lect. Notes Pure Appl. Math., Vol. 114, Dekker, 1989 ). MR 90c:58086 | Zbl 0676.58036 · Zbl 0676.58036  J. MILNOR , Dynamics in one Complex Variable : Introductory Lectures , Stony Brook I.M.S. Preprint 1990 / 1995 .  L. R. GOLDBERG , Fixed Points of Polynomial Maps , Part I, Rotation subsets of the circle. Numdam | Zbl 0771.30027 · Zbl 0771.30027 · numdam:ASENS_1992_4_25_6_679_0 · eudml:82332  C. PETERSEN , On the Pommerenke-Levin-Yoccoz inequality , preprint I.H.E.S., 1991 . · Zbl 0802.30022  A. POIRIER , On the Realization of Fixed Point Portraits , Stony Brook I.M.S., Preprint, 1991 / 2000 .  A. POIRIER , On Postcritically Finite Polynomials (Thesis, Stony Brook, in preparation). · Zbl 1179.37066  D. SULLIVAN , Conformal Dynamical Systems , pp. 725-752 of Geometric Dynamics, PALIS Ed., Lecture Notes Math., No. 1007, Springer, 1983 ). MR 85m:58112 | Zbl 0524.58024 · Zbl 0524.58024  W. THURSTON , On the Combinatorics of Iterated Rational Maps , preprint, Princeton University, circa, 1985 .  W. THURSTON , Three-dimensional Geometry and Topology (to appear).  P. VEERMAN , Symbolic dynamics of order preserving orbits, Physica , Vol. 29D, 1987 , pp. 191-201. MR 89h:58133 | Zbl 0625.28012 · Zbl 0625.28012 · doi:10.1016/0167-2789(87)90055-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.