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The full periodicity kernel of the trefoil. (English) Zbl 0834.54024
Summary: We consider the following topological spaces: ${\bold O}=\{z\in {\bbfC}:|z+i|=1\}, {\bold O}_3={\bold O}\cup\{z\in {\bbfC}:z^4\in[0,1],\text{Im} z\geq 0\}, {\bold O}_4={\bold O}\cup\{ z\in {\bbfC}:z^4\in[0,1]\}, \infty_1={\bold O}\cup\{z\in{\bbfC} :|z-i|=1\}\cup\{z\in{\bbfC} :z\in[0,1]\}, \infty_2=\infty_1\cup\{z\in{\bbfC} :z^2\in[0,1]\}$, and ${\bold T} =\{z\in{\bbfC} :z=\cos(3\theta)e^{i\theta}, 0\leq\theta\leq 2\pi\}$. Set $E\in\lbrace{\bold O}_3,{\bold O}_4,\infty_1,\infty_2, {\bold T}\}$. An $E$ map $f$ is a continuous self-map of $E$ having the branching point fixed. We denote by $\text{Per}(f)$ the set of periods of all periodic points of $f$. The set $K\subset{\bbfN}$ is the {\it full periodicity kernel} of $E$ if it satisfies the following two conditions: (1) If $f$ is an $E$ map and $K\subset \text{Per}(f)$, then $\text{Per} (f)={\bbfN}$. (2) If $S\subset{\bbfN}$ is a set such that for every $E$ map $f$, $S\subset\text{Per}(f)$ implies $\text{Per}(f)={\bbfN}$, then $K\subset S$. We compute the full periodicity kernel of ${\bold O}_3,{\bold O}_4,\infty_1,\infty_2$ and ${\bold T}$.
54H20Topological dynamics
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