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December  2017, 37(12): 6183-6188. doi: 10.3934/dcds.2017267

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

The authors are grateful to CONACYT for financial support CB-2012/181247 and CB-2015/255633 given to this work

Received  December 2016 Revised  July 2017 Published  August 2017

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.
Citation: Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267
References:
[1]

G. Blé, External arguments and invariant measures for the quadratic family, Disc. and Cont. Dyn. Sys., 11 (2004), 241-260.  doi: 10.3934/dcds.2004.11.241.  Google Scholar

[2]

A. Douady, Systémes dynamiques holomorphes, Astérisque, (1983), 105-106.   Google Scholar

[3]

A. Douady, Algorithm for computing angles in the Mandelbrot set, Chaotic Dynamics and Fractals, Atlanta 1985, Notes Rep. Math. Sci. Eng., 2 (1986), 155-168.   Google Scholar

[4]

A. Douady and J. H. Hubbard, Étude Dynamique Des Polynômes Complexes I et II, Département de Mathématiques, Orsay, 1985.  Google Scholar

[5]

J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175 (2009), 335-415.  doi: 10.1007/s00222-008-0152-8.  Google Scholar

[6]

J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: Three theorems of J.-C. Yoccoz, Topological Methods in Modern Mathematics, (1993), 467-511.   Google Scholar

[7]

M. Lyubich, Dynamics of quadratic polynomials Ⅰ-Ⅱ, Acta Math, 178 (1997), 185-297.  doi: 10.1007/BF02392694.  Google Scholar

[8]

M. Martens and T. Nowicki, Invariant measures for typical quadratic maps, Asterisque, 261 (2000), 239-252.   Google Scholar

[9]

J. Milnor, Dynamics in One Complex Variable, Introductory lectures. Friedr. Vieweg & Sohn, Braunschweig, 1999.  Google Scholar

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T. Nowicki and S. van Strien, Invariant measures exist under a summability condition for unimodal maps, Invent. Math., 105 (1991), 123-136.  doi: 10.1007/BF01232258.  Google Scholar

show all references

The authors are grateful to CONACYT for financial support CB-2012/181247 and CB-2015/255633 given to this work

References:
[1]

G. Blé, External arguments and invariant measures for the quadratic family, Disc. and Cont. Dyn. Sys., 11 (2004), 241-260.  doi: 10.3934/dcds.2004.11.241.  Google Scholar

[2]

A. Douady, Systémes dynamiques holomorphes, Astérisque, (1983), 105-106.   Google Scholar

[3]

A. Douady, Algorithm for computing angles in the Mandelbrot set, Chaotic Dynamics and Fractals, Atlanta 1985, Notes Rep. Math. Sci. Eng., 2 (1986), 155-168.   Google Scholar

[4]

A. Douady and J. H. Hubbard, Étude Dynamique Des Polynômes Complexes I et II, Département de Mathématiques, Orsay, 1985.  Google Scholar

[5]

J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175 (2009), 335-415.  doi: 10.1007/s00222-008-0152-8.  Google Scholar

[6]

J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: Three theorems of J.-C. Yoccoz, Topological Methods in Modern Mathematics, (1993), 467-511.   Google Scholar

[7]

M. Lyubich, Dynamics of quadratic polynomials Ⅰ-Ⅱ, Acta Math, 178 (1997), 185-297.  doi: 10.1007/BF02392694.  Google Scholar

[8]

M. Martens and T. Nowicki, Invariant measures for typical quadratic maps, Asterisque, 261 (2000), 239-252.   Google Scholar

[9]

J. Milnor, Dynamics in One Complex Variable, Introductory lectures. Friedr. Vieweg & Sohn, Braunschweig, 1999.  Google Scholar

[10]

T. Nowicki and S. van Strien, Invariant measures exist under a summability condition for unimodal maps, Invent. Math., 105 (1991), 123-136.  doi: 10.1007/BF01232258.  Google Scholar

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