Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-12T17:01:05.943Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalizations of Douady’s magic formula

Part of: Complex dynamical systems

Published online by Cambridge University Press:  28 July 2021

ADAM EPSTEIN  and
GIULIO TIOZZO Open the ORCID record for GIULIO TIOZZO [Opens in a new window]
ADAM EPSTEIN
Affiliation:
Mathematics Institute, University of Warwick, Coventry, UK (email: [email protected])
GIULIO TIOZZO*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada
*
email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We generalize a combinatorial formula of Douady from the main cardioid to other hyperbolic components H of the Mandelbrot set, constructing an explicit piecewise linear map which sends the set of angles of external rays landing on H to the set of angles of external rays landing on the real axis.

Keywords

Mandelbrot setexternal rayshyperbolic componentsmain cardioid

MSC classification

Primary: 37F20: Combinatorics and topology
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 9 , September 2022 , pp. 2784 - 2799
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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

Blé, G. and Cabrera, C.. A generalization of Douady’s formula. Discrete Contin. Dyn. Syst. 37(12) (2017), 6183–6188.CrossRefGoogle Scholar
Blé, G.. External arguments and invariant measures for the quadratic family. Discrete Contin. Dyn. Syst. 11(2–3) (2004), 241–260.CrossRefGoogle Scholar
Bullett, S. and Sentenac, P.. Ordered orbits of the shift, square roots, and the devil’s staircase. Math. Proc. Cambridge Philos. Soc. 115(3) (1994), 451–481.CrossRefGoogle Scholar
Douady, A.. Topological entropy of unimodal maps: monotonicity for quadratic polynomials. Real and Complex Dynamical Systems (Hillerød, 1993) (NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 464). Ed. Branner, B. and Hjorth, P.. Kluwer,Dordrecht, 1995, pp. 65–87.CrossRefGoogle Scholar
Shallit, J.. Automaticity IV: sequences, sets, and diversity. J. Théor. Nombres Bordeaux 8(2) (1996), 347–367.CrossRefGoogle Scholar
Thurston, W.. On the geometry and dynamics of iterated rational maps. Complex Dynamics. Ed. Schleicher, D. and Selinger, N.. A K Peters,Wellesley, MA, 2009, pp. 3–137.CrossRefGoogle Scholar
Zakeri, S.. External rays and the real slice of the Mandelbrot set. Ergod. Th. & Dynam. Sys. 23 (2003), 637–660.CrossRefGoogle Scholar