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

## A generalization of Douady's formula

 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.

Citation: Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267
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##### References:
 [1] Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451 [2] Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101 [3] Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185 [4] Gamaliel Blé. External arguments and invariant measures for the quadratic family. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 241-260. doi: 10.3934/dcds.2004.11.241 [5] Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35 [6] Adrian Tudorascu. On absolutely continuous curves of probabilities on the line. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5105-5124. doi: 10.3934/dcds.2019207 [7] Jagannathan Gomatam, Isobel McFarlane. Generalisation of the Mandelbrot set to integral functions of quaternions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 107-116. doi: 10.3934/dcds.1999.5.107 [8] Huaibin Li. An equivalent characterization of the summability condition for rational maps. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4567-4578. doi: 10.3934/dcds.2013.33.4567 [9] Lucia D. Simonelli. Absolutely continuous spectrum for parabolic flows/maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 263-292. doi: 10.3934/dcds.2018013 [10] Rogelio Valdez. Self-similarity of the Mandelbrot set for real essentially bounded combinatorics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (4) : 897-922. doi: 10.3934/dcds.2006.16.897 [11] James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667 [12] Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365 [13] Oliver Jenkinson. Optimization and majorization of invariant measures. Electronic Research Announcements, 2007, 13: 1-12. [14] Siniša Slijepčević. Stability of invariant measures. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1345-1363. doi: 10.3934/dcds.2009.24.1345 [15] Moises Delgado, Heeralal Janwa. Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: The gold case. Advances in Mathematics of Communications, 2017, 11 (2) : 389-396. doi: 10.3934/amc.2017033 [16] Héctor A. Tabares-Ospina, Mauricio Osorio. Methodology for the characterization of the electrical power demand curve, by means of fractal orbit diagrams on the complex plane of Mandelbrot set. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1895-1905. doi: 10.3934/dcdsb.2020008 [17] Zhihong Xia. Hyperbolic invariant sets with positive measures. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 811-818. doi: 10.3934/dcds.2006.15.811 [18] Marcus Pivato. Invariant measures for bipermutative cellular automata. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 723-736. doi: 10.3934/dcds.2005.12.723 [19] Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 [20] Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975

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