A generalization of Douady's formula

Gamaliel Blé; Carlos Cabrera

doi:10.3934/dcds.2017267

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2017, Volume 37, Issue 12: 6183-6188. Doi: 10.3934/dcds.2017267

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A generalization of Douady's formula

Gamaliel Blé^1, , and
Carlos Cabrera^2,

1.
División Académica de Ciencias Básicas, UJAT, Km. 1, Carretera Cunduacán-Jalpa de Méndez, C.P. 86690, Cunduacán Tabasco, México
2.
Instituto de Matemáticas de la UNAM, Unidad Cuernavaca, Av. Universidad s/n. Col. Lomas de Chamilpa, C.P. 62210, Cuernavaca, Morelos, México

* Corresponding author: Gamaliel Blé
* Corresponding author: Gamaliel Blé

Received: November 30, 2016

Revised: June 30, 2017

Published: October 2017

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Abstract

The Douady's formula was defined for the external argument on the boundary points of the main hyperbolic component $W_0$ of the Mandelbrot set $M$ and it is given by the map $T(θ)=1/2+θ/4$ . We extend this formula to the boundary of all hyperbolic components of $M$ and we give a characterization of the parameter in $M$ with these external arguments.

Keywords:

Mandelbrot set,
quadratic polynomials,
tuning map,
summability condition,
absolutely continuous invariant measures.

Mathematics Subject Classification: Primary: 37F10, 37F45; Secondary: 37F50.

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References

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