The full periodicity kernel of the trefoil

Leseduarte, Carme; Llibre, Jaume

doi:10.5802/aif.1512

Leseduarte, Carme ; Llibre, Jaume

Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 219-262.

Résumé

We consider the following topological spaces: $O = {z \in ℂ : | z + i | = 1}$ , $O_{3} = O \cup {z \in ℂ : z^{4} \in [0, 1], Im z \geq 0}$ , $O_{4} = O \cup {z \in ℂ : z^{4} \in [0, 1]}$ , $\infty_{1} = O : | z - i | = 1} \cup {z \in ℂ : z \in [0, 1]}$ , $\infty_{2} = \infty_{1} \cup {z \in ℂ : z^{2} \in [0, 1]}$ , et $T = {z \in ℂ : z = \cos (3 θ) e^{i θ}, 0 \leq θ \leq 2 π}$ . Set $E \in {O_{3}, O_{4}, \infty_{1}, \infty_{2}, T}$ . An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $Per (f)$ 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 $K \subset Per (f)$ , then $Per (f) = ℕ$ . (2) If $S \subset ℕ$ is a set such that for every $E$ map $f$ , $S \subset Per (f)$ implies $Per (f) = ℕ$ , then $K \subset S$ . In this paper we compute the full periodicity kernel of $O_{3}, O_{4}, \infty_{1}, \infty_{2}$ and $T$ .

EuDML MR Zbl

DOI : 10.5802/aif.1512

@article{AIF_1996__46_1_219_0,
     author = {Leseduarte, Carme and Llibre, Jaume},
     title = {The full periodicity kernel of the trefoil},
     journal = {Annales de l'Institut Fourier},
     pages = {219--262},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {1},
     year = {1996},
     doi = {10.5802/aif.1512},
     mrnumber = {1385516},
     zbl = {0834.54024},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1512/}
}

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Leseduarte, Carme; Llibre, Jaume. The full periodicity kernel of the trefoil. Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 219-262. doi : 10.5802/aif.1512. http://www.numdam.org/articles/10.5802/aif.1512/

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