Strong bifurcation loci of full Hausdorff dimension
[Lieux de bifurcation maximale de dimension de Hausdorff totale]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 6, pp. 947-984.

Dans l’espace des modules d des fractions rationnelles de degré d, le lieu de bifurcation est le support d’un (1,1)-courant positif fermé T bif qui est appelé courant de bifurcation. Ce courant induit une mesure μ bif :=(T bif ) 2d-2 dont le support est le siège de bifurcations maximales. Notre principal résultat stipule que supp (μ bif ) est de dimension de Hausdorff maximale 2(2d-2). Par conséquent, l’ensemble des fractions rationnelles de degré d possédant (2d-2) cycles neutres distincts est dense dans un ensemble de dimension de Hausdorff totale.

In the moduli space d of degree d rational maps, the bifurcation locus is the support of a closed (1,1) positive current T bif which is called the bifurcation current. This current gives rise to a measure μ bif :=(T bif ) 2d-2 whose support is the seat of strong bifurcations. Our main result says that supp (μ bif ) has maximal Hausdorff dimension 2(2d-2). As a consequence, the set of degree d rational maps having (2d-2) distinct neutral cycles is dense in a set of full Hausdorff dimension.

DOI : 10.24033/asens.2181
Classification : 37F45, 32U15, 28A78
Keywords: complex dynamics, bifurcations, pluripotential theory, Hausdorff dimension
Mot clés : dynamique holomorphe, bifurcations, théorie du pluripotentiel, dimension de Hausdorff
@article{ASENS_2012_4_45_6_947_0,
     author = {Gauthier, Thomas},
     title = {Strong bifurcation loci of full {Hausdorff} dimension},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {947--984},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {6},
     year = {2012},
     doi = {10.24033/asens.2181},
     mrnumber = {3075109},
     language = {en},
     url = {http://archive.numdam.org/articles/10.24033/asens.2181/}
}
TY  - JOUR
AU  - Gauthier, Thomas
TI  - Strong bifurcation loci of full Hausdorff dimension
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2012
SP  - 947
EP  - 984
VL  - 45
IS  - 6
PB  - Société mathématique de France
UR  - http://archive.numdam.org/articles/10.24033/asens.2181/
DO  - 10.24033/asens.2181
LA  - en
ID  - ASENS_2012_4_45_6_947_0
ER  - 
%0 Journal Article
%A Gauthier, Thomas
%T Strong bifurcation loci of full Hausdorff dimension
%J Annales scientifiques de l'École Normale Supérieure
%D 2012
%P 947-984
%V 45
%N 6
%I Société mathématique de France
%U http://archive.numdam.org/articles/10.24033/asens.2181/
%R 10.24033/asens.2181
%G en
%F ASENS_2012_4_45_6_947_0
Gauthier, Thomas. Strong bifurcation loci of full Hausdorff dimension. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 6, pp. 947-984. doi : 10.24033/asens.2181. http://archive.numdam.org/articles/10.24033/asens.2181/

[1] M. Aspenberg, Rational Misiurewicz maps are rare, Comm. Math. Phys. 291 (2009), 645-658. | MR | Zbl

[2] M. Aspenberg & J. Graczyk, Dimension and measure for semi-hyperbolic rational maps of degree 2, C. R. Math. Acad. Sci. Paris 347 (2009), 395-400. | MR | Zbl

[3] G. Bassanelli & F. Berteloot, Bifurcation currents in holomorphic dynamics on k , J. reine angew. Math. 608 (2007), 201-235. | MR | Zbl

[4] G. Bassanelli & F. Berteloot, Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Math. Ann. 345 (2009), 1-23. | MR | Zbl

[5] G. Bassanelli & F. Berteloot, Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J. 201 (2011), 23-43. | MR | Zbl

[6] E. Bedford & B. A. Taylor, The Dirichlet problem for a complex Monge-Ampere equation, Bull. Amer. Math. Soc. 82 (1976), 102-104. | MR | Zbl

[7] L. Bers & H. L. Royden, Holomorphic families of injections, Acta Math. 157 (1986), 259-286. | MR | Zbl

[8] F. Berteloot, C. Dupont & L. Molino, Normalization of bundle holomorphic contractions and applications to dynamics, Ann. Inst. Fourier (Grenoble) 58 (2008), 2137-2168. | Numdam | MR | Zbl

[9] F. Berteloot & V. Mayer, Rudiments de dynamique holomorphe, Cours Spécialisés 7, Soc. Math. France, 2001. | MR | Zbl

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

[11] X. Buff & A. L. Epstein, Bifurcation measure and postcritically finite rational maps, in Complex dynamics : families and friends / edited by Dierk Schleicher, A K Peters, Ltd., 2009, 491-512. | MR | Zbl

[12] E. M. Chirka, Complex analytic sets, Mathematics and its Applications (Soviet Series) 46, Kluwer Academic Publishers Group, 1989. | MR | Zbl

[13] L. Demarco, Dynamics of rational maps: a current on the bifurcation locus, Math. Res. Lett. 8 (2001), 57-66. | MR | Zbl

[14] L. Demarco, Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326 (2003), 43-73. | MR | Zbl

[15] T.-C. Dinh & N. Sibony, Dynamics in several complex variables: endomorphisms of projective spaces and polynomial-like mappings, in Holomorphic dynamical systems, Lecture Notes in Math. 1998, Springer, 2010, 165-294. | MR | Zbl

[16] R. Dujardin, Approximation des fonctions lisses sur certaines laminations, Indiana Univ. Math. J. 55 (2006), 579-592. | MR | Zbl

[17] R. Dujardin, Cubic polynomials: a measurable view on parameter space, in Complex dynamics : families and friends / edited by Dierk Schleicher, A K Peters, Ltd., 2009, 451-490. | MR | Zbl

[18] R. Dujardin & C. Favre, Distribution of rational maps with a preperiodic critical point, Amer. J. Math. 130 (2008), 979-1032. | MR | Zbl

[19] R. Mañé, P. Sad & D. P. Sullivan, On the dynamics of rational maps, Ann. Sci. École Norm. Sup. 16 (1983), 193-217. | Numdam | MR | Zbl

[20] P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Math. 44, Cambridge Univ. Press, 1995. | MR | Zbl

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

[22] C. T. Mcmullen, Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), 535-593. | MR | Zbl

[23] C. T. Mcmullen, The Mandelbrot set is universal, in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser. 274, Cambridge Univ. Press, 2000, 1-17. | MR | Zbl

[24] C. T. Mcmullen & D. P. Sullivan, Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), 351-395. | MR | Zbl

[25] W. De Melo & S. Van Strien, One-dimensional dynamics, Ergebn. Math. Grenzg. 25, Springer, 1993. | MR | Zbl

[26] J. Milnor, On Lattès maps, in Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, 9-43. | MR | Zbl

[27] J. Rivera-Letelier, On the continuity of Hausdorff dimension of Julia sets and similarity between the Mandelbrot set and Julia sets, Fund. Math. 170 (2001), 287-317. | MR | Zbl

[28] W. Rudin, Real and complex analysis, third éd., McGraw-Hill Book Co., 1987. | MR | Zbl

[29] M. Shishikura, The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. 147 (1998), 225-267. | MR | Zbl

[30] M. Shishikura & L. Tan, An alternative proof of Mañé's theorem on non-expanding Julia sets, in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser. 274, Cambridge Univ. Press, 2000, 265-279. | MR | Zbl

[31] N. Sibony, Dynamique des applications rationnelles de 𝐏 k , in Dynamique et géométrie complexes (Lyon, 1997), Panor. & Synthèses 8, Soc. Math. France, 1999. | MR | Zbl

[32] J. H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Math. 241, Springer, 2007. | MR | Zbl

[33] S. Van Strien, Misiurewicz maps unfold generically (even if they are critically non-finite), Fund. Math. 163 (2000), 39-54. | MR | Zbl

[34] L. Tan, Hausdorff dimension of subsets of the parameter space for families of rational maps. (A generalization of Shishikura's result), Nonlinearity 11 (1998), 233-246. | MR | Zbl

[35] M. Urbański, Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414. | MR | Zbl

Cité par Sources :