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$\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem
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 |
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.
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. |
| [2] |
A. Douady,
Systémes dynamiques holomorphes, Astérisque, (1983), 105-106.
|
| [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.
|
| [4] |
A. Douady and J. H. Hubbard, Étude Dynamique Des Polynômes Complexes I et II, Département de Mathématiques, Orsay, 1985. |
| [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. |
| [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.
|
| [7] |
M. Lyubich,
Dynamics of quadratic polynomials Ⅰ-Ⅱ, Acta Math, 178 (1997), 185-297.
doi: 10.1007/BF02392694. |
| [8] |
M. Martens and T. Nowicki,
Invariant measures for typical quadratic maps, Asterisque, 261 (2000), 239-252.
|
| [9] |
J. Milnor, Dynamics in One Complex Variable, Introductory lectures. Friedr. Vieweg & Sohn, Braunschweig, 1999. |
| [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. |
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. |
| [2] |
A. Douady,
Systémes dynamiques holomorphes, Astérisque, (1983), 105-106.
|
| [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.
|
| [4] |
A. Douady and J. H. Hubbard, Étude Dynamique Des Polynômes Complexes I et II, Département de Mathématiques, Orsay, 1985. |
| [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. |
| [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.
|
| [7] |
M. Lyubich,
Dynamics of quadratic polynomials Ⅰ-Ⅱ, Acta Math, 178 (1997), 185-297.
doi: 10.1007/BF02392694. |
| [8] |
M. Martens and T. Nowicki,
Invariant measures for typical quadratic maps, Asterisque, 261 (2000), 239-252.
|
| [9] |
J. Milnor, Dynamics in One Complex Variable, Introductory lectures. Friedr. Vieweg & Sohn, Braunschweig, 1999. |
| [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. |
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