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