In the moduli space of degree rational maps, the bifurcation locus is the support of a closed positive current which is called the bifurcation current. This current gives rise to a measure whose support is the seat of strong bifurcations. Our main result says that has maximal Hausdorff dimension . As a consequence, the set of degree rational maps having distinct neutral cycles is dense in a set of full Hausdorff dimension.
Dans l’espace des modules des fractions rationnelles de degré , le lieu de bifurcation est le support d’un -courant positif fermé qui est appelé courant de bifurcation. Ce courant induit une mesure dont le support est le siège de bifurcations maximales. Notre principal résultat stipule que est de dimension de Hausdorff maximale . Par conséquent, l’ensemble des fractions rationnelles de degré possédant cycles neutres distincts est dense dans un ensemble de dimension de Hausdorff totale.
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://www.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://www.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://www.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, Serie 4, Volume 45 (2012) no. 6, pp. 947-984. doi : 10.24033/asens.2181. http://www.numdam.org/articles/10.24033/asens.2181/
[1] Rational Misiurewicz maps are rare, Comm. Math. Phys. 291 (2009), 645-658. | MR | Zbl
,[2] Dimension and measure for semi-hyperbolic rational maps of degree 2, C. R. Math. Acad. Sci. Paris 347 (2009), 395-400. | MR | Zbl
& ,[3] Bifurcation currents in holomorphic dynamics on , J. reine angew. Math. 608 (2007), 201-235. | MR | Zbl
& ,[4] Lyapunov exponents, bifurcation currents and laminations in bifurcation loci, Math. Ann. 345 (2009), 1-23. | MR | Zbl
& ,[5] Distribution of polynomials with cycles of a given multiplier, Nagoya Math. J. 201 (2011), 23-43. | MR | Zbl
& ,[6] The Dirichlet problem for a complex Monge-Ampere equation, Bull. Amer. Math. Soc. 82 (1976), 102-104. | MR | Zbl
& ,[7] Holomorphic families of injections, Acta Math. 157 (1986), 259-286. | MR | Zbl
& ,[8] Normalization of bundle holomorphic contractions and applications to dynamics, Ann. Inst. Fourier (Grenoble) 58 (2008), 2137-2168. | Numdam | MR | Zbl
, & ,[9] Rudiments de dynamique holomorphe, Cours Spécialisés 7, Soc. Math. France, 2001. | MR | Zbl
& ,[10] The iteration of cubic polynomials. I. The global topology of parameter space, Acta Math. 160 (1988), 143-206. | MR | Zbl
& ,[11] 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] Complex analytic sets, Mathematics and its Applications (Soviet Series) 46, Kluwer Academic Publishers Group, 1989. | MR | Zbl
,[13] Dynamics of rational maps: a current on the bifurcation locus, Math. Res. Lett. 8 (2001), 57-66. | MR | Zbl
,[14] Dynamics of rational maps: Lyapunov exponents, bifurcations, and capacity, Math. Ann. 326 (2003), 43-73. | MR | Zbl
,[15] 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] Approximation des fonctions lisses sur certaines laminations, Indiana Univ. Math. J. 55 (2006), 579-592. | MR | Zbl
,[17] 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] Distribution of rational maps with a preperiodic critical point, Amer. J. Math. 130 (2008), 979-1032. | MR | Zbl
& ,[19] On the dynamics of rational maps, Ann. Sci. École Norm. Sup. 16 (1983), 193-217. | Numdam | MR | Zbl
, & ,[20] Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Math. 44, Cambridge Univ. Press, 1995. | MR | Zbl
,[21] Complex dynamics and renormalization, Annals of Math. Studies 135, Princeton Univ. Press, 1994. | MR | Zbl
,[22] Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), 535-593. | MR | Zbl
,[23] 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] Quasiconformal homeomorphisms and dynamics. III. The Teichmüller space of a holomorphic dynamical system, Adv. Math. 135 (1998), 351-395. | MR | Zbl
& ,[25] One-dimensional dynamics, Ergebn. Math. Grenzg. 25, Springer, 1993. | MR | Zbl
& ,[26] On Lattès maps, in Dynamics on the Riemann sphere, Eur. Math. Soc., Zürich, 2006, 9-43. | MR | Zbl
,[27] 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] Real and complex analysis, third éd., McGraw-Hill Book Co., 1987. | MR | Zbl
,[29] The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets, Ann. of Math. 147 (1998), 225-267. | MR | Zbl
,[30] 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] Dynamique des applications rationnelles de , in Dynamique et géométrie complexes (Lyon, 1997), Panor. & Synthèses 8, Soc. Math. France, 1999. | MR | Zbl
,[32] The arithmetic of dynamical systems, Graduate Texts in Math. 241, Springer, 2007. | MR | Zbl
,[33] Misiurewicz maps unfold generically (even if they are critically non-finite), Fund. Math. 163 (2000), 39-54. | MR | Zbl
,[34] 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] Rational functions with no recurrent critical points, Ergodic Theory Dynam. Systems 14 (1994), 391-414. | MR | Zbl
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