Trees and the dynamics of polynomials

DeMarco, Laura G.; McMullen, Curtis T.

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 3, pp. 337-383.

Dans ce travail, nous étudions des revêtements ramifiés d’arbres métriques simpliciaux $F:T\to T$ qui sont obtenus à partir d’applications polynomiales $f:ℂ\to ℂ$ possédant un ensemble de Julia non connexe. Nous montrons que la collection de tous ces arbres, à un facteur d’échelle près, forme un espace contractile $\mathbb{P}{T}_D$ qui compactifie l’espace des modules des polynômes de degré $D$. Nous montrons aussi que $F$ enregistre le comportement asymptotique des multiplicateurs de $f$ et que toute famille méromorphe de polynômes définis sur ${\Delta }^{*}$ peut être complétée par un unique arbre comme sa fibre centrale. Dans le cas cubique, nous donnons une énumération combinatoire des arbres ainsi obtenus et montrons que $\mathbb{P}{T}_3$ est lui-même un arbre.

In this paper we study branched coverings of metrized, simplicial trees $F:T\to T$ which arise from polynomial maps $f:ℂ\to ℂ$ with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space $\mathbb{P}{T}_D$ compactifying the moduli space of polynomials of degree $D$; that $F$ records the asymptotic behavior of the multipliers of $f$; and that any meromorphic family of polynomials over ${\Delta }^{*}$ can be completed by a unique tree at its central fiber. In the cubic case we give a combinatorial enumeration of the trees that arise, and show that $\mathbb{P}{T}_3$ is itself a tree.

[1] M. Baker & R. Rumely, Potential theory on the Berkovich projective line, in preparation.

[2] R. L. Benedetto, Reduction, dynamics, and Julia sets of rational functions, J. Number Theory 86 (2001), 175-195. | MR | Zbl

[3] B. Branner & J. H. Hubbard, The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math. 160 (1988), 143-206. | Zbl

[4] B. Branner & J. H. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169 (1992), 229-325. | Zbl

[5] H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103-144 (1965). | MR | Zbl

[6] A. L. Edmonds, R. S. Kulkarni & R. E. Stong, Realizability of branched coverings of surfaces, Trans. Amer. Math. Soc. 282 (1984), 773-790. | Zbl

[7] N. D. Emerson, Dynamics of polynomials with disconnected Julia sets, Discrete Contin. Dyn. Syst. 9 (2003), 801-834. | MR | Zbl

[8] N. D. Emerson, Brownian motion, random walks on trees, and harmonic measure on polynomial Julia sets, preprint, 2006.

[9] C. Favre & J. Rivera-Letelier, Théorie ergodique des fractions rationnelles sur un corps ultramétrique, preprint, 2007. | Zbl

[10] A. Freire, A. Lopes & R. Mañé, An invariant measure for rational maps, Bol. Soc. Brasil. Mat. 14 (1983), 45-62. | Zbl

[11] M. Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques 1, CEDIC, 1981. | MR | Zbl

[12] M. Gromov, On the entropy of holomorphic maps, Enseign. Math. 49 (2003), 217-235. | MR | Zbl

[13] D. M. Harris, Turning curves for critically recurrent cubic polynomials, Nonlinearity 12 (1999), 411-418. | MR | Zbl

[14] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, in Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, 1993, 467-511. | MR | Zbl

[15] J. Kiwi, Puiseux series polynomial dynamics and iteration of complex cubic polynomials, Ann. Inst. Fourier (Grenoble) 56 (2006), 1337-1404. | Numdam | MR | Zbl

[16] O. Kozlovski & S. Van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, preprint, 2006. | Zbl

[17] O. Lehto & K. I. Virtanen, Quasiconformal mappings in the plane, second éd., Springer, 1973. | Zbl

[18] M. J. Lyubich, Entropy properties of rational endomorphisms of the Riemann sphere, Ergodic Theory Dynam. Systems 3 (1983), 351-385. | MR | Zbl

[19] C. T. Mcmullen, Automorphisms of rational maps, in Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ. 10, Springer, 1988, 31-60. | MR | Zbl

[20] C. T. Mcmullen, The classification of conformal dynamical systems, in Current developments in mathematics, 1995 (Cambridge, MA), 323-360, Int. Press, Cambridge, MA, 1994. | MR | Zbl

[21] C. T. Mcmullen, Complex dynamics and renormalization, Annals of Mathematics Studies 135, Princeton University Press, 1994. | MR | Zbl

[22] C. T. Mcmullen, Ribbon $ℝ$-trees and holomorphic dynamics on the unit disk, preprint, 2007. | MR | Zbl

[23] J. Milnor, Local connectivity of Julia sets: expository lectures, in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser. 274, Cambridge Univ. Press, 2000, 67-116. | MR | Zbl

[24] J. W. Morgan, Trees and hyperbolic geometry, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., 1987, 590-597. | MR | Zbl

[25] J. W. Morgan & P. B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I, Ann. of Math. 120 (1984), 401-476. | Zbl

[26] J. W. Morgan & P. B. Shalen, An introduction to compactifying spaces of hyperbolic structures by actions on trees, in Geometry and topology (College Park, Md., 1983/84), Lecture Notes in Math. 1167, Springer, 1985, 228-240. | Zbl

[27] J.-P. Otal, Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3, Astérisque 235 (1996). | Zbl

[28] W. Parry, Entropy and generators in ergodic theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. | MR | Zbl

[29] F. Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels, Invent. Math. 94 (1988), 53-80. | MR | Zbl

[30] R. Pérez-Marco, Fixed points and circle maps, Acta Math. 179 (1997), 243-294. | MR | Zbl

[31] K. M. Pilgrim, Dessins d'enfants and Hubbard trees, Ann. Sci. École Norm. Sup. 33 (2000), 671-693. | MR | Zbl

[32] F. Przytycki & J. Skrzypczak, Convergence and pre-images of limit points for coding trees for iterations of holomorphic maps, Math. Ann. 290 (1991), 425-440. | Zbl

[33] W. Qiu & Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets, preprint, 2006. | Zbl

[34] J. Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, in Geometric methods in dynamics. II, Astérisque 287, 2003, 147-230. | Numdam | MR | Zbl

[35] J. Rivera-Letelier, Points périodiques des fonctions rationnelles dans l’espace hyperbolique $p$-adique, Comment. Math. Helv. 80 (2005), 593-629. | MR | Zbl

[36] L. Sario & M. Nakai, Classification theory of Riemann surfaces, Die Grund. Math. Wiss., Band 164, Springer, 1970. | Zbl

[37] M. Shishikura, Trees associated with the configuration of Herman rings, Ergodic Theory Dynam. Systems 9 (1989), 543-560. | MR | Zbl

[38] G. Springer, Riemann surfaces, Chelsea Publishing Co., 1981.

[39] R. Vakil, Genus 0 and 1 Hurwitz numbers: recursions, formulas, and graph-theoretic interpretations, Trans. Amer. Math. Soc. 353 (2001), 4025-4038. | MR | Zbl