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Semi-parabolic tools for hyperbolic Hénon maps and continuity of Julia sets in $ \mathbb{C}^{2}$

Authors: Remus Radu and Raluca Tanase
Journal: Trans. Amer. Math. Soc. 370 (2018), 3949-3996
MSC (2010): Primary 37F45, 37D99, 32A99, 47H10
Published electronically: December 18, 2017
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Abstract: We prove some new continuity results for the Julia sets $ J$ and $ J^{+}$ of the complex Hénon map $ H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $ a$ and $ c$ are complex parameters. We look at the parameter space of dissipative Hénon maps which have a fixed point with one eigenvalue $ (1+t)\lambda $, where $ \lambda $ is a root of unity and $ t$ is real and small in absolute value. These maps have a semi-parabolic fixed point when $ t$ is 0, and we use the techniques that we have developed in a prior work for the semi-parabolic case to describe nearby perturbations. We show that for small nonzero $ \vert t\vert$, the Hénon map is hyperbolic and has connected Julia set. We prove that the Julia sets $ J$ and $ J^{+}$ depend continuously on the parameters as $ t\rightarrow 0$, which is a two-dimensional analogue of radial convergence from one-dimensional dynamics. Moreover, we prove that this family of Hénon maps is stable on $ J$ and $ J^{+}$ when $ t$ is non-negative.

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  • [A] Zin Arai, On hyperbolic plateaus of the Hénon map, Experiment. Math. 16 (2007), no. 2, 181-188. MR 2339274
  • [AI] Z. Arai and Y. Ishii, On parameter loci of the Hénon family, arXiv:1501.01368v2
  • [B] Eric Bedford, Complex Hénon maps with semi-parabolic fixed points, J. Difference Equ. Appl. 16 (2010), no. 5-6, 425-426. MR 2642457,
  • [BD] P. Berger and R. Dujardin, On stability and hyperbolicity for polynomial automorphisms of $ \mathbb{C}^{2}$, arXiv:1409.4449, to appear in Ann. École Norm. Sup.
  • [BH] X. Buff and J. H. Hubbard, Dynamics in One Complex Variable, to appear.
  • [Br] Felix E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968), 27-35. MR 0230180
  • [BS] Eric Bedford and John Smillie, Real polynomial diffeomorphisms with maximal entropy: Tangencies, Ann. of Math. (2) 160 (2004), no. 1, 1-26. MR 2119716,
  • [BS1] Eric Bedford and John Smillie, Polynomial diffeomorphisms of $ {\bf C}^2$: currents, equilibrium measure and hyperbolicity, Invent. Math. 103 (1991), no. 1, 69-99. MR 1079840,
  • [BS6] Eric Bedford and John Smillie, Polynomial diffeomorphisms of $ {\bf C}^2$. VI. Connectivity of $ J$, Ann. of Math. (2) 148 (1998), no. 2, 695-735. MR 1668567,
  • [BS7] Eric Bedford and John Smillie, Polynomial diffeomorphisms of $ {\bf C}^2$. VII. Hyperbolicity and external rays, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 4, 455-497 (English, with English and French summaries). MR 1693587,
  • [BSU17] Eric Bedford, John Smillie, and Tetsuo Ueda, Semi-parabolic bifurcations in complex dimension two, Comm. Math. Phys. 350 (2017), no. 1, 1-29. MR 3606468
  • [D] Adrien Douady, Does a Julia set depend continuously on the polynomial?, Complex dynamical systems (Cincinnati, OH, 1994) Proc. Sympos. Appl. Math., vol. 49, Amer. Math. Soc., Providence, RI, 1994, pp. 91-138. MR 1315535,
  • [DH] A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR 762431
  • [DL] Romain Dujardin and Mikhail Lyubich, Stability and bifurcations for dissipative polynomial automorphisms of $ \mathbb{C}^2$, Invent. Math. 200 (2015), no. 2, 439-511. MR 3338008,
  • [FM] Shmuel Friedland and John Milnor, Dynamical properties of plane polynomial automorphisms, Ergodic Theory Dynam. Systems 9 (1989), no. 1, 67-99. MR 991490,
  • [FS] John Erik Fornæss and Nessim Sibony, Complex Hénon mappings in $ {\bf C}^2$ and Fatou-Bieberbach domains, Duke Math. J. 65 (1992), no. 2, 345-380. MR 1150591,
  • [H] J. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics. Volume 2: Surface Homeomorphisms and Rational Functions. Matrix Editions, Ithaca, NY (2016). MR 3675959
  • [Ha] Monique Hakim, Attracting domains for semi-attractive transformations of $ {\bf C}^p$, Publ. Mat. 38 (1994), no. 2, 479-499. MR 1316642,
  • [Haï] P. Haïssinsky, Applications de la chirurgie holomorphe aux systèmes dynamiques, notamment aux points paraboliques, Thèse de l'Université de Paris-Sud, Orsay, 1998.
  • [HOV1] John H. Hubbard and Ralph W. Oberste-Vorth, Hénon mappings in the complex domain. I. The global topology of dynamical space, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 5-46. MR 1307296
  • [HOV2] John H. Hubbard and Ralph W. Oberste-Vorth, Hénon mappings in the complex domain. II. Projective and inductive limits of polynomials, Real and complex dynamical systems (Hillerød, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 464, Kluwer Acad. Publ., Dordrecht, 1995, pp. 89-132. MR 1351520
  • [KH] Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374
  • [KS] William A. Kirk and Brailey Sims (eds.), Handbook of metric fixed point theory, Kluwer Academic Publishers, Dordrecht, 2001. MR 1904271
  • [Kw] Tomoki Kawahira, Semiconjugacies between the Julia sets of geometrically finite rational maps, Ergodic Theory Dynam. Systems 23 (2003), no. 4, 1125-1152. MR 1997970,
  • [L] P. Lavaurs, Systèmes dynamiques holomorphiques: explosion de points périodiques, Thèse, Université Paris-Sud, 1989.
  • [Mc] Curtis T. McMullen, Hausdorff dimension and conformal dynamics. II. Geometrically finite rational maps, Comment. Math. Helv. 75 (2000), no. 4, 535-593. MR 1789177,
  • [Mi] John Milnor, Dynamics in one complex variable, 3rd ed., Annals of Mathematics Studies, vol. 160, Princeton University Press, Princeton, NJ, 2006. MR 2193309
  • [MNTU] S. Morosawa, Y. Nishimura, M. Taniguchi, and T. Ueda, Holomorphic dynamics, Cambridge Studies in Advanced Mathematics, vol. 66, Cambridge University Press, Cambridge, 2000. Translated from the 1995 Japanese original and revised by the authors. MR 1747010
  • [R] Remus Andrei Radu, Topological models for hyperbolic and semi-parabolic complex Hénon maps. Pro-Quest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)-Cornell University. MR 3193181
  • [RT] Remus Radu and Raluca Tanase, A structure theorem for semi-parabolic Hénon maps, arXiv:1411.3824v1
  • [S] Michael Shub, Global stability of dynamical systems, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR 869255
  • [T] Raluca Elena Tanase, Henon maps, discrete groups and continuity of Julia sets, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)-Cornell University. MR 3193182
  • [Th] William P. Thurston, On the geometry and dynamics of iterated rational maps, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 3-137. Edited by Dierk Schleicher and Nikita Selinger and with an appendix by Schleicher. MR 2508255,
  • [U] Tetsuo Ueda, Local structure of analytic transformations of two complex variables. I, J. Math. Kyoto Univ. 26 (1986), no. 2, 233-261. MR 849219,

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Additional Information

Remus Radu
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660

Raluca Tanase
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660

Received by editor(s): September 3, 2015
Received by editor(s) in revised form: July 22, 2016, and September 6, 2016
Published electronically: December 18, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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