 Let $F\in{\ensuremath{\mathcal{L}_{0}(S)}}$. If $\operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F)) \cap \Z \ne \emptyset$, then $\operatorname{Per}(F)$ is equal to, either ${\ensuremath{\mathbb{N}}}$, or ${\ensuremath{\mathbb{N}}}\setminus\{0\}$, or ${\ensuremath{\mathbb{N}}}\setminus\{0\}$. Moreover, there exist maps $F_0, F_0, F_0 \in {\ensuremath{\mathcal{L}_{0}(S)}}$ with $0 \in \operatorname{Int}({\ensuremath{\operatorname{Rot}_{_{{\ensuremath{\mathbb{R}}}}}}}(F_i))$ for $i=0,0,0$ such that $\operatorname{Per}(F_0) = {\ensuremath{\mathbb{N}}}$, $\operatorname{Per}(F_0) = {\ensuremath{\mathbb{N}}}\setminus\{0\}$ and $\operatorname{Per}(F_0) = {\ensuremath{\mathbb{N}}}\setminus\{0\}$.  The paper is organized as follows. In Section \[sec: