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Chapter 6 Polynomials and external rays In this chapter we discuss polynomials and the combinatorial topol­ ogy of the Julia set. This material is in preparation for §7, where we will use renormalization to break the Julia set of a quadratic polyno­ mial into many connected pieces. These pieces can potentially touch at periodic cycles, so here we study the way in which the Julia set is connected at its periodic points. 6.1 Accessibility D efin ition s. Let K be a full nondegenerate continuum in the com­ plex plane. This means K is a compact connected set of

Abstract

We study conditions involving the critical set of a regular polynomial endomorphism f∶ℂ2↦ℂ2 under which all complete external rays from infinity for f have well defined endpoints.

In Celebration of John Milnor's 80th Birthday (PMS-51)

A N D E L B R O T s e t E(D) exceptional set Σ symbol space H{X) space of compact subsets Rt external ray

. The Sullivan Classification of Fatou Components . . . . . Using the Fatou Set to Study the Julia Set 17. Prime Ends and Local Connectivity . . . . . . 18. Polynomial Dynamics: External Rays 19. Hyperbolic and Subhyperbolic Maps . . . . . . . . . . Appendix A. Theorems from Classical Analysis . . . . Appendix B. Length-Area-Modulus Inequalities . . . . . Appendix C. Rotations, Continued Fractions, and Rational Approximation Appendix D. Two or More Complex Variables . . . . Appendix E. Branched Coverings and Orbifolds Appendix F. No Wandering Fatou Components Appendix

Index a fixed point 88 a-lamination 130 attracting 36 (3 fixed point 88 Caratheodory topology 66 C( f ) 67 collar 22 critically finite 38 crossed renormalization 111 curve system 190 cusp 24 cycle 36 disk 66 ergodic 42 essential 11 Euler characteristic x(C?) 180 external ray 84 Fatou set 36 Feigenbaum polynomial 114 filled Julia set 38 full continuum 83 7n(i) 135 hyperbolic metric 11 hyperbolic rational map 45 indifferent 36 infinitely renormalizable 121 J( f ) 36 Julia set 36 K ( f ), polynomial-like 71 K( f ) 38 Koebe principle 15

Bestimmung der „external rays" (ein Analogon zu elektrischen Feld- linien) und über eine Vielzahl von offenen Fragen und Problemen. Wer sich mit diesem Gebiet neu befas- sen will, findet hier eine Reihe von Ansatzpunkten. Gert Eilenberger steuert in sei- nem Beitrag einige philosophische Überlegungen zum Determinismus, über Freiheit und zur Möglichkeit der Abbildung realer Strukturen durch mathematische Strukturen bei. Er vertritt dabei den Standpunkt von Konrad Lorenz, wonach das Denken der Menschen in mathema- tischen Strukturen Ergebnis des Evolutionsprozesses ist

-Weierstrass Theorem, 129 centre, 81 Centre Problem, 82 Classification Theorem, 55 complete invariance, 28 conjugation, 24 continued fraction, 82 Cremer cycle, 26 Cremer point, 84 critical limit set, 84 critical point, 27 critical value, 27 cross cut, 7 cycle, 26 cycle of domains, 47 degree, 24 dendrite, 141 Denjoy-Wolff Theorem, 43 diophantine condition, 90 Dirac measure, 147 Douady-Hubbard Theorem, 160 equicontinuity, 3 equilibrium, 17 exceptional point, 30 exceptional set, 30 expanding mapping, 118 external ray, 150 Fatou domain, 54 Fatou set, 24

the Late Middle ages, under the influence of robert grosseteste (1175–1253) and roger bacon (1214–1294), extramission was gradually synthesized with and then replaced by the intromission theory of seeing.32 intro- mission is an assimilation process, in which the eye is regarded as a passive receptor of light and the viewer has a passive role in the act of seeing. in the intromission process, external rays emit- ted from objects imprint their impression on the merely receptive eye. it is a process whereby the external visible forms emanating from the

noColor January 6, 2014 7x10 Unmating of rational maps: Sufficient criteria and examples Daniel Meyer ABSTRACT. Douady and Hubbard introduced the operation of mating of polyno- mials. This identifies two filled Julia sets and the dynamics on them via external rays. In many cases one obtains a rational map. Here the opposite question is tack- led. Namely, we ask when a given (postcritically finite) rational map f arises as a mating. A sufficient condition when this is possible is given. If this condition is satisfied, we present a simple explicit algorithm to