Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Cubic polynomial maps with periodic critical orbit, Part II: Escape regions

Authors: Araceli Bonifant, Jan Kiwi and John Milnor
Journal: Conform. Geom. Dyn. 14 (2010), 68-112
MSC (2010): Primary 37F10, 30C10, 30D05
Published electronically: March 9, 2010
Erratum: Conform. Geom. Dyn. 14 (2010), 190-193.
MathSciNet review: 2600536
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The parameter space $ \mathcal{S}_p$ for monic centered cubic polynomial maps with a marked critical point of period $ p$ is a smooth affine algebraic curve whose genus increases rapidly with $ p$. Each $ \mathcal{S}_p$ consists of a compact connectedness locus together with finitely many escape regions, each of which is biholomorphic to a punctured disk and is characterized by an essentially unique Puiseux series. This note will describe the topology of $ \mathcal{S}_p$, and of its smooth compactification, in terms of these escape regions. In particular, it computes the Euler characteristic. It concludes with a discussion of the real sub-locus of $ \mathcal{S}_p$.

References [Enhancements On Off] (What's this?)

  • [AC] D. A. Aruliah and R. M. Corless, Numerical parameterization of affine varieties using ODE's, International Conference on Symbolic and Algebraic Computation. Proceedings 2004 International Symposium on Symbolic and Algebraic Computation, Santander, Spain, 2004, 12-18. MR 2126919
  • [Br] B. Branner, Cubic polynomials, turning around the connectedness locus, pp. 391-427 of ``Topological Methods in Mathematics'' (edit. Goldberg and Phillips), Publish or Perish, 1993. MR 1215972 (94c:58168)
  • [BM] A. Bonifant and J. Milnor, Cubic polynomial maps with periodic critical orbit, Part III: External rays, in preparation.
  • [BH] B. Branner and J. H. Hubbard, The iteration of cubic polynomials II, patterns and parapatterns, Acta Math. 169 (1992) 229-325. MR 1194004 (94d:30044)
  • [DMS] L. DeMarco and A. Schiff, Enumerating the basins of infinity for cubic polynomials. To appear, Special Volume of Journal of Difference Equations and Applications (2010).
  • [H] D. Harris, Turning curves for critically recurrent cubic polynomials, Nonlinearity 12 2 (1999), 411-418. MR 1677771 (2000a:37028)
  • [He] C. Heckman, Monotonicity and the construction of quasiconformal conjugacies in the real cubic family, Thesis, Stony Brook 1996.
  • [IK] H. Inou and J. Kiwi, Combinatorics and topology of straightening maps I: compactness and bijectivity, ArXiv:0809.1262.
  • [IKR] H. Inou, J. Kiwi and P. Roesch, work in preparation.
  • [K1] J. Kiwi, Puiseux series polynomial dynamics and iteration of complex cubic polynomials, Ann. Inst. Fourier (Grenoble) 56 (2006) 1337-1404. MR 2273859 (2007h:37066)
  • [K2] J. Kiwi, manuscript in preparation.
  • [M] J. Milnor, Cubic Polynomial Maps with Periodic Critical Orbit, Part I, In: ``Complex Dynamics Families and Friends'', ed. D. Schleicher, A. K. Peters 2009, pp. 333-411. MR 2508263
  • [MTr] J. Milnor and C. Tresser, On entropy and monotonicity for real cubic maps, Comm. Math. Phys. 209 (2000) 123-178. MR 1736945 (2001e:37048)
  • [MTh] J. Milnor and W. Thurston, On iterated maps of the interval, In: ``Dynamical systems'', Alexander, J.C. (ed.). Lecture Notes in Mathematics N1342. Berlin: Springer, 1988, pp. 465-563. MR 970571 (90a:58083)
  • [R] J. Rivera-Letelier, Dynamique des fonctions rationnelles sur des corps locaux, Geometric methods in dynamics. II. Astérisque No. 287 (2003), xv, 147-230. MR 2040006 (2005f:37100)

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2010): 37F10, 30C10, 30D05

Retrieve articles in all journals with MSC (2010): 37F10, 30C10, 30D05

Additional Information

Araceli Bonifant
Affiliation: Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881

Jan Kiwi
Affiliation: Facultad de Matemáticas, Pontificia Universidad Católica, Casilla 306, Correo 22, Santiago de Chile, Chile

John Milnor
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660

Received by editor(s): September 3, 2009
Published electronically: March 9, 2010
Additional Notes: The first author was partially supported by the Simons Foundation
The second author was supported by Research Network on Low Dimensional Dynamics PBCT/CONICYT, Chile
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society