The full periodicity kernel of the trefoil

Leseduarte, Carme ; Llibre, Jaume

Annales de l'Institut Fourier, Tome 46 (1996), p. 219-262 / Harvested from Numdam

Résumé

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 ${\bf T}$.

Détail

Publié le : 1996-01-01

MR 1385516 | Zbl 0834.54024

DOI : https://doi.org/10.5802/aif.1512

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