American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$
  • DCDS Home
  • This Issue
  • Next Article
    $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem
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 - A, 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

[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

show all references

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

[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]

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
[3]

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
[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]

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
[17]

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
[18]

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
[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]

Piotr Kościelniak, Marcin Mazur, Piotr Oprocha, Paweł Pilarczyk. Shadowing is generic---a continuous map case. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3591-3609. doi: 10.3934/dcds.2014.34.3591

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (20)
  • HTML views (26)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences