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October  2017, 37(10): 5085-5104. doi: 10.3934/dcds.2017220

On parabolic external maps

Luna Lomonaco 1,, , Carsten Lunde Petersen 2, and  Weixiao Shen 3,

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

* Corresponding author

Received  March 2016 Revised  April 2017 Published  June 2017

Fund Project: The first author has been supported by FAPESP via the process 2013/20480-7. The second author has been supported by the Danish Council for Independent Research | Natural Sciences via the grant DFF -4181-00502.

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.

Keywords: Circle covering, metric expanding, smooth conjugacy, parabolic-like map, parabolic external map.
Mathematics Subject Classification: Primary: 37E10; Secondary: 37F15.
Citation: Luna Lomonaco, Carsten Lunde Petersen, Weixiao Shen. On parabolic external maps. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5085-5104. doi: 10.3934/dcds.2017220
References:
[1]

B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge University Press, 2014.  Google Scholar

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G. Cui, Circle expanding maps and symmetric structures, Ergodic Theory and Dynamical Systems, 18 (1998), 831-842.  doi: 10.1017/S0143385798117455.  Google Scholar

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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.  Google Scholar

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L. Lomonaco, Parabolic-like maps, Ergodic Theory and Dynamical Systems, 35 (2015), 2171-2197.  doi: 10.1017/etds.2014.27.  Google Scholar

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J. Ma., On Evolution of a Class of Markov Maps, Undergraduate thesis (in Chinese), University of Science and Technology of China, 2007. Google Scholar

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R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Communications in Mathematical Physics, 100 (1985), 495-524.  doi: 10.1007/BF01217727.  Google Scholar

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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.  Google Scholar

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W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, 1993. doi: 10.1007/BF02392981.  Google Scholar

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W. Rudin, Real and Complex Analysis, New York-Toronto, Ont. -London, 1966.  Google Scholar

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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.   Google Scholar

show all references

References:
[1]

B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge University Press, 2014.  Google Scholar

[2]

G. Cui, Circle expanding maps and symmetric structures, Ergodic Theory and Dynamical Systems, 18 (1998), 831-842.  doi: 10.1017/S0143385798117455.  Google Scholar

[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.  Google Scholar

[4]

L. Lomonaco, Parabolic-like maps, Ergodic Theory and Dynamical Systems, 35 (2015), 2171-2197.  doi: 10.1017/etds.2014.27.  Google Scholar

[5]

J. Ma., On Evolution of a Class of Markov Maps, Undergraduate thesis (in Chinese), University of Science and Technology of China, 2007. Google Scholar

[6]

R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Communications in Mathematical Physics, 100 (1985), 495-524.  doi: 10.1007/BF01217727.  Google Scholar

[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.  Google Scholar

[8]

W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer-Verlag, 1993. doi: 10.1007/BF02392981.  Google Scholar

[9]

W. Rudin, Real and Complex Analysis, New York-Toronto, Ont. -London, 1966.  Google Scholar

[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.   Google Scholar

Figure 1.  A map of the maps we consider. $\mathcal{F}_d^{1+\text{BV}}$ is the set of degree $d$ smooth covering $h:\mathcal{S}^1\to\mathcal{S}^1$ with $h\in C^{1+\text{BV}}$; $\mathcal{O}_d^{1+\text{BV}}$ is the set of maps $h\in\mathcal{F}_d^{1+\text{BV}}$ for which for every periodic point $p$ say of period $s$, there is a neighborhood $U(p)$ of $p$ such that for all $x\in U(p)\setminus \{p\}$ we have $Dh^s(x)>1$; while $\mathcal{M}_d^{1+\text{BV}}$ and $\mathcal{T}_d^{1+\text{BV}}$ are the class of respectively metrically and topologically expanding $h\in F_d^{1+\text{BV}}$. $\mathcal{F}_d$ is the class of real analytic degree $d$ circle coverings, $\mathcal{T}_d$ and $\mathcal{M}_d$ the set of respectively topologically and metrically expanding $h \in \mathcal{F}_d$, and $\mathcal{T}_{d,*}$ and $\mathcal{M}_{d,*}$ the set of respectively topologically and metrically expanding $h \in \mathcal{F}_d$ for which $\text{Par}(h) \neq \emptyset$. Also, $\mathcal{P}_d$ is the class of extenal maps and $\mathcal{P}_{d,*}$ the class of parabolic external maps. Finally, $\mathcal{H}_{d,1} =\{ h \in \mathcal{F}_d | \,\,h \sim_{qs} h_d (z)= \frac{z^d+(d-1)/(d+1)}{(d-1)z^d/(d+1)+1}\}$. By Corollary 2.2, $\mathcal{O}_d^{1+\text{BV}}=\mathcal{T}_d^{1+\text{BV}}$, and by Theorem 2.4, $\mathcal{M}_d\subset\mathcal{P}_d=\mathcal{T}_{d}$, $\mathcal{M}_{d,*}~\subset~\mathcal{P}_{d,*}~=~\mathcal{T}_{d,*}$ and $\mathcal{M}_{d,1}\subset\mathcal{P}_{d,1}=\mathcal{H}_{d,1}=\mathcal{T}_{d,1}$.
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Figure 2.  A parabolic external map in $\mathcal{P}_{d,1}$.
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Figure 3.  Construction
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