# The full periodicity kernel of the trefoil

• Volume: 46, Issue: 1, page 219-262
• ISSN: 0373-0956

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## Abstract

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We consider the following topological spaces: $\mathbf{O}=\left\{z\in ℂ:|z+i|=1\right\}$, ${\mathbf{O}}_{3}=\mathbf{O}\cup \left\{z\in ℂ:{z}^{4}\in \left[0,1\right],Im\phantom{\rule{0.166667em}{0ex}}z\ge 0\right\}$, ${\mathbf{O}}_{4}=\mathbf{O}\cup \left\{z\in ℂ:{z}^{4}\in \left[0,1\right]\right\}$, ${\infty }_{1}=\mathbf{O}:|z-i|=1\right\}\cup \left\{z\in ℂ:z\in \left[0,1\right]\right\}$, ${\infty }_{2}={\infty }_{1}\cup \left\{z\in ℂ:{z}^{2}\in \left[0,1\right]\right\}$, et $\mathbf{T}=\left\{z\in ℂ:z=\mathrm{cos}\left(3\theta \right){e}^{i\theta },\phantom{\rule{3.33333pt}{0ex}}0\le \theta \le 2\pi \right\}$. Set $E\in \left\{{\mathbf{O}}_{3},{\mathbf{O}}_{4},{\infty }_{1},{\infty }_{2},\mathbf{T}\right\}$. An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $Per\left(f\right)$ the set of periods of all periodic points of $f$. The set $K\subset ℕ$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) If $f$ is an $E$ map and $K\subset Per\left(f\right)$, then $Per\left(f\right)=ℕ$. (2) If $S\subset ℕ$ is a set such that for every $E$ map $f$, $S\subset Per\left(f\right)$ implies $Per\left(f\right)=ℕ$, then $K\subset S$. In this paper we compute the full periodicity kernel of ${\mathbf{O}}_{3},{\mathbf{O}}_{4},{\infty }_{1},{\infty }_{2}$ and $\mathbf{T}$.

## How to cite

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Leseduarte, Carme, and Llibre, Jaume. "The full periodicity kernel of the trefoil." Annales de l'institut Fourier 46.1 (1996): 219-262. <http://eudml.org/doc/75172>.

@article{Leseduarte1996,
abstract = {We consider the following topological spaces: $\mathbf\{O\}=\lbrace z\in \{\Bbb C\}:\vert z+i\vert =1\rbrace$, $\mathbf\{O\}_3=\mathbf\{O\}\cup \lbrace z\in \{\Bbb C\}:z^4\in [0,1],\operatorname\{Im\}\,z\ge 0\rbrace$, $\mathbf\{O\}_4=\mathbf\{O\}\cup \lbrace z\in \{\Bbb C\}:z^4\in [0,1]\rbrace$, $\infty _1=\mathbf\{O\} :\vert z-i\vert =1\rbrace \cup \lbrace z\in \{\Bbb C\} :z\in [0,1]\rbrace$, $\infty _2=\infty _1\cup \lbrace z\in \{\Bbb C\} :z^2\in [0,1]\rbrace$, et $\mathbf\{T\} =\lbrace z\in \{\Bbb C\} :z=\{\rm cos\}(3\theta )e^\{i\theta \},~0\le \theta \le 2\pi \rbrace$. Set $E\in \lbrace \mathbf\{O\}_3,\mathbf\{O\}_4,\infty _1,\infty _2, \mathbf\{T\}\rbrace$. An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $\operatorname\{Per\}(f)$ the set of periods of all periodic points of $f$. The set $K\subset \{\Bbb N\}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) If $f$ is an $E$ map and $K\subset \operatorname\{Per\}(f)$, then $\operatorname\{Per\} (f)=\{\Bbb N\}$. (2) If $S\subset \{\Bbb N\}$ is a set such that for every $E$ map $f$, $S\subset \operatorname\{Per\}(f)$ implies $\operatorname\{Per\}(f)=\{\Bbb N\}$, then $K\subset S$. In this paper we compute the full periodicity kernel of $\{\bf O\}_3,\{\bf O\}_4,\infty _1,\infty _2$ and $\{\bf T\}$.},
author = {Leseduarte, Carme, Llibre, Jaume},
journal = {Annales de l'institut Fourier},
keywords = {periods; full periodicity kernel},
language = {eng},
number = {1},
pages = {219-262},
publisher = {Association des Annales de l'Institut Fourier},
title = {The full periodicity kernel of the trefoil},
url = {http://eudml.org/doc/75172},
volume = {46},
year = {1996},
}

TY - JOUR
AU - Leseduarte, Carme
AU - Llibre, Jaume
TI - The full periodicity kernel of the trefoil
JO - Annales de l'institut Fourier
PY - 1996
PB - Association des Annales de l'Institut Fourier
VL - 46
IS - 1
SP - 219
EP - 262
AB - We consider the following topological spaces: $\mathbf{O}=\lbrace z\in {\Bbb C}:\vert z+i\vert =1\rbrace$, $\mathbf{O}_3=\mathbf{O}\cup \lbrace z\in {\Bbb C}:z^4\in [0,1],\operatorname{Im}\,z\ge 0\rbrace$, $\mathbf{O}_4=\mathbf{O}\cup \lbrace z\in {\Bbb C}:z^4\in [0,1]\rbrace$, $\infty _1=\mathbf{O} :\vert z-i\vert =1\rbrace \cup \lbrace z\in {\Bbb C} :z\in [0,1]\rbrace$, $\infty _2=\infty _1\cup \lbrace z\in {\Bbb C} :z^2\in [0,1]\rbrace$, et $\mathbf{T} =\lbrace z\in {\Bbb C} :z={\rm cos}(3\theta )e^{i\theta },~0\le \theta \le 2\pi \rbrace$. Set $E\in \lbrace \mathbf{O}_3,\mathbf{O}_4,\infty _1,\infty _2, \mathbf{T}\rbrace$. An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $\operatorname{Per}(f)$ the set of periods of all periodic points of $f$. The set $K\subset {\Bbb N}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) If $f$ is an $E$ map and $K\subset \operatorname{Per}(f)$, then $\operatorname{Per} (f)={\Bbb N}$. (2) If $S\subset {\Bbb N}$ is a set such that for every $E$ map $f$, $S\subset \operatorname{Per}(f)$ implies $\operatorname{Per}(f)={\Bbb N}$, then $K\subset S$. In this paper we compute the full periodicity kernel of ${\bf O}_3,{\bf O}_4,\infty _1,\infty _2$ and ${\bf T}$.
LA - eng
KW - periods; full periodicity kernel
UR - http://eudml.org/doc/75172
ER -

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