# ANNALES DE L'INSTITUT FOURIER

The full periodicity kernel of the trefoil
Annales de l'Institut Fourier, Volume 46 (1996) no. 1, pp. 219-262.

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 $\mathbf{T}$.

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title = {The full periodicity kernel of the trefoil},
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Leseduarte, Carme; Llibre, Jaume. The full periodicity kernel of the trefoil. Annales de l'Institut Fourier, Volume 46 (1996) no. 1, pp. 219-262. doi : 10.5802/aif.1512. https://aif.centre-mersenne.org/articles/10.5802/aif.1512/

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