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On parabolic external maps
1. | Departamento de Matemática Aplicada, Instituto de Matemática e Estatistica, Universidade de São Paulo, Rua do Matão 1010,05508-090 São Paulo -SP, Brazil |
2. | Department of Science, NSM, IMFUFA, Roskilde University, Universitetsvej 1, 4000 Roskilde, Denmark |
3. | Shanghai Center for Mathematical Sciences and School of Mathematical Sciences, Fudan University, Handan Road 220, Shanghai, China 200433 |
We prove that any $C^{1+\text{BV}}$ degree d ≥ 2 circle covering $h$ having all periodic orbits weakly expanding, is conjugate by a $C^{1+\text{BV}}$ diffeomorphism to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.
References:
[1] |
B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge University Press, 2014. |
[2] |
G. Cui,
Circle expanding maps and symmetric structures, Ergodic Theory and Dynamical Systems, 18 (1998), 831-842.
doi: 10.1017/S0143385798117455. |
[3] |
A. Douady and J. H. Hubbard,
On the dynamics of polynomial-like mappings, Annales scientifiques de l'École normale supérieure, 18 (1985), 287-343.
doi: 10.24033/asens.1491. |
[4] |
L. Lomonaco,
Parabolic-like maps, Ergodic Theory and Dynamical Systems, 35 (2015), 2171-2197.
doi: 10.1017/etds.2014.27. |
[5] |
J. Ma., On Evolution of a Class of Markov Maps, Undergraduate thesis (in Chinese), University of Science and Technology of China, 2007. |
[6] |
R. Mañé,
Hyperbolicity, sinks and measure in one-dimensional dynamics, Communications in Mathematical Physics, 100 (1985), 495-524.
doi: 10.1007/BF01217727. |
[7] |
M. Martens, W. de Melo and S. van Strien,
Julia-Fatou-Sullivan theory for real onedimensional dynamics, Acta Mathematica, 168 (1992), 273-318.
doi: 10.1007/BF02392981. |
[8] |
W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, 1993.
doi: 10.1007/BF02392981. |
[9] |
W. Rudin, Real and Complex Analysis, New York-Toronto, Ont. -London, 1966. |
[10] |
M. Shishikura,
Bifurcation of parabolic fixed points, in The Mandelbrot set, theme and variations, London Mathematical Society Lecture Note Series, Cambridge University Press, 274 (2000), 325-363.
|
show all references
References:
[1] |
B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge University Press, 2014. |
[2] |
G. Cui,
Circle expanding maps and symmetric structures, Ergodic Theory and Dynamical Systems, 18 (1998), 831-842.
doi: 10.1017/S0143385798117455. |
[3] |
A. Douady and J. H. Hubbard,
On the dynamics of polynomial-like mappings, Annales scientifiques de l'École normale supérieure, 18 (1985), 287-343.
doi: 10.24033/asens.1491. |
[4] |
L. Lomonaco,
Parabolic-like maps, Ergodic Theory and Dynamical Systems, 35 (2015), 2171-2197.
doi: 10.1017/etds.2014.27. |
[5] |
J. Ma., On Evolution of a Class of Markov Maps, Undergraduate thesis (in Chinese), University of Science and Technology of China, 2007. |
[6] |
R. Mañé,
Hyperbolicity, sinks and measure in one-dimensional dynamics, Communications in Mathematical Physics, 100 (1985), 495-524.
doi: 10.1007/BF01217727. |
[7] |
M. Martens, W. de Melo and S. van Strien,
Julia-Fatou-Sullivan theory for real onedimensional dynamics, Acta Mathematica, 168 (1992), 273-318.
doi: 10.1007/BF02392981. |
[8] |
W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, 1993.
doi: 10.1007/BF02392981. |
[9] |
W. Rudin, Real and Complex Analysis, New York-Toronto, Ont. -London, 1966. |
[10] |
M. Shishikura,
Bifurcation of parabolic fixed points, in The Mandelbrot set, theme and variations, London Mathematical Society Lecture Note Series, Cambridge University Press, 274 (2000), 325-363.
|



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