Hostname: page-component-848d4c4894-tn8tq Total loading time: 0 Render date: 2024-06-28T00:34:52.695Z Has data issue: false hasContentIssue false
  • English
  • Français

The parameter space of cubic laminations with a fixed critical leaf

Published online by Cambridge University Press:  26 July 2016

ALEXANDER BLOKH ,
LEX OVERSTEEGEN ,
ROSS PTACEK  and
VLADLEN TIMORIN
ALEXANDER BLOKH
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA email ablokh@math.uab.edu, overstee@math.uab.edu
LEX OVERSTEEGEN
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, USA email ablokh@math.uab.edu, overstee@math.uab.edu
ROSS PTACEK
Affiliation:
Faculty of Mathematics, National Research University Higher School of Economics, Vavilova St. 7, 112312 Moscow, Russia email rptacek@uab.edu
VLADLEN TIMORIN
Affiliation:
Faculty of Mathematics, National Research University Higher School of Economics, Vavilova St. 7, 112312 Moscow, Russia email rptacek@uab.edu Independent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia email vtimorin@hse.ru
Get access Rights & Permissions [Opens in a new window]

Abstract

Thurston parameterized quadratic invariant laminations with a non-invariant lamination, the quotient of which yields a combinatorial model for the Mandelbrot set. As a step toward generalizing this construction to cubic polynomials, we consider slices of the family of cubic invariant laminations defined by a fixed critical leaf with non-periodic endpoints. We parameterize each slice by a lamination just as in the quadratic case, relying on the techniques of smart criticality previously developed by the authors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 8 , December 2017 , pp. 2453 - 2486
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blokh, A. and Levin, G.. Growing trees, laminations and the dynamics on the Julia set. Ergod. Th. & Dynam. Sys. 22 (2002), 63–97.CrossRefGoogle Scholar
Blokh, A., Mimbs, D., Oversteegen, L. and Valkenburg, K.. Laminations in the language of leaves. Trans. Amer. Math. Soc. 365 (2013), 5367–5391.CrossRefGoogle Scholar
Blokh, A., Oversteegen, L., Ptacek, R. and Timorin, V.. Combinatorial models for spaces of cubic polynomials. Preprint, 2014, arXiv:1405.4287.Google Scholar
Cosper, D., Houghton, J., Mayer, J., Mernik, L. and Olson, J.. Central strips of sibling leaves in laminations of the unit disk. Top. Proc. 48 (2016), 69–100.Google Scholar
Kiwi, J.. Wandering orbit portraits. Trans. Amer. Math. Soc. 354 (2002), 1473–1485.Google Scholar
Milnor, J.. Dynamics in One Complex Variable (Annals of Mathematical Studies, 160) . Princeton University Press, Princeton, NJ, 2006.Google Scholar
Schleicher, D.. On fibers and local connectivity of Mandelbrot and Multibrot sets. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot (Proceedings of Symposia in Pure Mathematics, 72) . Eds. Lapidus, M. and van Frankenhuysen, M.. American Mathematical Society, Providence, RI, 2004, pp. 477–517, Part 1.CrossRefGoogle Scholar
Schleicher, D.. Appendix: laminations, Julia sets, and the Mandelbrot set. Complex Dynamics: Families and Friends. Ed. Schleicher, D.. A K Peters, Wellesley, MA, 2009, pp. 111–130.Google Scholar
Thurston, W.. The combinatorics of iterated rational maps. Complex Dynamics: Families and Friends. Ed. Schleicher, D.. A K Peters, Wellesley, MA, 2009, pp. 1–108.Google Scholar