American Institute of Mathematical Sciences

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

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

* 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.

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
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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
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 $\mathcal{M}_{d,1}\subset\mathcal{P}_{d,1}=\mathcal{H}_{d,1}=\mathcal{T}_{d,1}$.
A parabolic external map in $\mathcal{P}_{d,1}$.
Construction
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