A zero topological entropy map with recurrent points not $F_\sigma$

We show that there is a continuous map $\chi$ of the unit interval into itself of type $2^\infty$ which has a trajectory disjoint from the set $\operatorname{Rec}(\chi )$ of recurrent points of $\chi$, but contained in the closure of $\operatorname{Rec}(\chi )$. In particular, $\operatorname{Rec}(\chi )$ is not closed. A function $\psi$ of type $2^\infty$, with nonclosed set of recurrent points, was found by H. Chu and J. Xiong [Proc. Amer. Math. Soc. 97 (1986), 361-366]. However, there is no trajectory contained in $\overline {\operatorname{Rec} (\psi)}\setminus \operatorname{Rec}(\psi)$, since any point in $\overline { \operatorname{Rec}(\psi)}$ is eventually mapped into $\operatorname{Rec} (\psi)$. Moreover, our construction is simpler.

We use $\chi$ to show that there is a continuous map of the interval of type $2^\infty$ for which the set of recurrent points is not an $F_\sigma$set. This example disproves a conjecture of A. N. Sharkovsky et al., from 1989. We also provide another application of $\chi$.