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Original Article

Elliptic Curves as Attractors in ℙ2 Part 1: Dynamics

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Pages 385-420 | Published online: 30 Jan 2011

  • L. V. Ahlfors. Lectures on Quasiconformal Mappings. Princeton: Van Nostrand, 1966.
  • J. C. Alexander, I. Kan, J. Yorke, and Z. You. "Riddled Basins." Int. J. Bifurcation and Chaos 2 (1992), 795–813.
  • M. Artebani and I. Dolgachev. "The Hesse Pencil of Plane Cubic Curves." arXiv:math.AG/0611590, 2006.
  • P. Ashwin, J. Buescu, and I. Stewart. "From Attractor to Chaotic Saddle: A Tale of Transverse Instability." Nonlinearity 9 (1996), 703–737.
  • P. Ashwin, P. Aston, and M. Nicol. "On the Unfolding of a Blowout Bifurcation." Physica D 111 (1998), 81–95.
  • J. Auslander, N. P. Bhatia, and P. Seibert. "Attractors in Dynamical Systems." Bol. Soc. Mat. Mexicana (2) 9 (1964), 55–66.
  • A. Avila, X. Buff, and A. Cheritat. "Siegel Disks with Smooth Boundaries." Acta Math. 193 (2004), 1–30.
  • E. Bedford and J. Smillie. "Polynomial Diffeomorphisms of C2. II: Stable Manifolds and Recurrence." J. Amer. Math. Soc. 4:4 (1991), 657–679.
  • A. Bonifant and M. Dabija. "Self- Maps of P2 with Invariant Elliptic Curves." In Complex Manifolds and Hyperbolic Geometry (Guanajuato, 2001), pp. 1–25, Contemp. Math. 311. Providence, RI: Amer. Math. Soc., 2002.
  • J.-Y. Briend and J. Duval. "Exposants de Lyapounoff et distribution des points périodiques répulsifs d'un endomorphisme de CPk." Acta Math. 182 (1999), 143–157.
  • J.-Y. Briend and J. Duval. "Deux caract érisations de la mesure d'équilibre d'un endomorphisme de Pk(C)." Publ. Math. Inst. Hautes Études Sci. 93 (2001), 145–159.
  • W.-L. Chow "On Compact Complex Analytic Varieties." Amer. J. Math. 71 (1949), 893–914.
  • A. Desboves. "Résolution en nombres entiers et sous sa forme la plus générale de l'équation cubique, homeg`ene á trois inconnues." Nouv. Ann. de la Math. Ser. III 5 (1886), 545–579.
  • N. Fakhruddin. "Questions on Self Maps of Algebraic Varieties." J. Ramanujan Math. Soc. 18:2 (2003), 109–122.
  • J. E. Fornæss. Dynamics in Several Complex Variables, CBMS Regional Conference Series in Mathematics, 87. Providence, RI: Amer. Math. Soc., 1996.
  • J. E. Fornæss and N. Sibony. "Complex Dynamics in Higher Dimension I." Astérisque 222 (1994), 201–231.
  • J. E. Fornæss and N. Sibony. "Classification of Recurrent Domains for Some Holomorphic Maps." Math. Ann. 301:4 (1995), 813–820.
  • J. E. Fornæss and N. Sibony. "Complex Dynamics in Higher Dimension II." In Modern Methods in Complex Analysis, edited by T. Bloom, D. Catlin, J. P. D'Angelo, and Y.-T. Siu, pp. 135–182, Ann. of Math. Stud., 137. Princeton, NJ: Princeton University Press, 1995.
  • J. E. Fornæss and N. Sibony. "Oka's Inequality for Currents and Applications." Math. Ann. 301:3 (1995), 399–419.
  • J. E. Fornæss and N. Sibony. "Dynamics of P2 (Examples)." In Laminations and Foliations in Dynamics, Geometry and Topology, pp. 47–85, Contemp. Math. 269. Providence, RI: Amer. Math. Soc., 2001.
  • J. E. Fornæss and B. Weickert. "Attractors in P2." In Several Complex Variables, pp. 297–307, Math. Sci. Res. Inst. Publ. 37. Cambridge, UK: Cambridge Univ. Press, 1999.
  • É. Ghys. "Transformations holomorphes au voisinage d'une courbe de Jordan" C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), 385–388.
  • A. Gorodetski and Y. Ilyashenko. "Minimal and Strange Attractors." Int. J. Bifurcations and Chaos 6 (1996), 1177–1183.
  • P. Griffiths and H. Harris. Principles of Algebraic Geometry. New York: Wiley, 1994.
  • V. Guedj. "Ergodic Properties of Rational Mappings with Large Topological Degree." Ann. of Math. 161 (2005), 1589–1607.
  • O. Hesse. " Über die Elimination der Variabeln aus drei algebraischen Gleichungen vom zweiten Grade mit zwei Variabeln." Crelle's J. 28 (1844), 68–96.
  • J. H. Hubbard and P. Papadopol. "Superattractive Fixed Points in Cn." Indiana Univ. Math. J. 43:1 (1994), 321–365.
  • M. Jonsson and B. Weickert. "A Nonalgebraic Attractor in P2." Proc. Amer. Math. Soc. 128:10 (2000), 2999–3002.
  • I. Kan. "Open Sets of Diffeomorphisms Having Two Attractors, Each with an Everywhere Dense Basin." Bull. Amer. Math. Soc. 31 (1994), 68–74.
  • K. Kaneko. "Dominance of Milnor Attractors in Globally Coupled Dynamical Systems with More Than 7†2 Degrees of Freedom." Phys. Rev. E. 66 (2002), 055201.
  • "Prevalence of Milnor Attractors and Chaotic Itinerancy in 'High'-Dimensional Dynamical Systems." In Synchronization: Theory and Application,, pp. 65–77, NATO Sci. Ser. II Math. Phys. Chem., 109. Dordrecht, The Netherlands: Kluwer Acad. Publ., 2003.
  • S. L. Krushkal_. Quasiconformal Mappings and Riemann Surfaces. New York: John Wiley and Sons, 1979.
  • Y. Maistrenko, V. Maistrenko, and A. Popovich. "Transverse Instability and Riddled Basins in a System of Two Coupled Logistic Maps." Phys. Rev. E 57 (1998), 2713–2724.
  • W. de Melo and S. van Strien. One-Dimensional Dynamics. New York: Springer-Verlag, 1993.
  • J. Milnor. "On the Concept of Attractor." Comm. Math. Phys. 99 (1985), 177–195; "Correction and Remarks." Comm. Math. Phys. 102, 517–519.
  • J. Milnor. "On Latt`es Maps." In Dynamics on the Riemann Sphere, A Bodil Branner Festschrift, edited by P. Hjorth and C. L. Petersen. European Math. Soc., 2006.
  • J. Milnor. Dynamics in One Complex Variable, Annals of Math. Study 130. Princeton, NJ: Princeton University Press, 2006.
  • E. Ott, J. Sommerer, J. Alexander, I. Kan, and J. Yorke. "Scaling Behavior of Chaotic Systems with Riddled Basins." Phys. Rev. Lett. 71 (1993), 4134–4137.
  • E. Ott and J. Sommerer. "Blowout Bifurcations: The Occurrence of Riddled Basins and On- Off Indeterminacy." Phys. Lett. A 188 (1994), 39–47.
  • N. Platt, E. Spiegel, and C. Tresser. "On- Off Intermittency: A Mechanism for Bursting." Phys. Rev. Lett. 70 (1993), 279–282.
  • M. Shishikura. "On the Quasiconformal Surgery of Rational Functions." Ann. Sci. École Norm. Sup.(4)20:1 (1987), 1–29.
  • N. Sibony. "Dynamique des applications rationnelles de Pk." Dynamique et Geometrie Complexes, Panoramas et Syntheses 8 (1999), 97–185.
  • T. Ueda. "Complex Dynamical Systems on Projective Spaces." In Chaotic Dynamical Systems, pp. 120–138. River Edge, NJ: World Scientif. Publ., 1993.
  • J.-C. Yoccoz. "Analytic Linearization of Circle Diffeomorphisms." In Dynamical Systems and Small Divisors (Cetraro, 1998), pp. 125–173, Lecture Notes in Math. 1784. Berlin: Springer, 2002.

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