On the $\omega$-limit sets of tent maps

Andrew D. Barwell ¹   ; Gareth Davies ²   ; Chris Good ³

¹ School of Mathematics University of Bristol Howard House Queens Avenue Bristol, BS8 1SN, UK and School of Mathematics University of Birmingham Birmingham, B15 2TT, UK
² Mathematical Institute University of Oxford 24-29 St. Giles' Oxford, OX1 3LB, UK
³ School of Mathematics University of Birmingham Birmingham, B15 2TT, UK

Fundamenta Mathematicae, Tome 217 (2012) no. 1, pp. 35-54

Résumé

For a continuous map $f$ on a compact metric space $(X,d)$, a set $D\subset X$ is internally chain transitive if for every $x,y\in D$ and every $\delta>0$ there is a sequence of points $\langle x=x_0,x_1,\ldots,x_n=y\rangle$ such that $d(f(x_i),x_{i+1}) \delta$ for $0\leq i n$. In this paper, we prove that for tent maps with periodic critical point, every closed, internally chain transitive set is necessarily an $\omega$-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an $\omega$-limit set. Together, these results lead us to conjecture that for tent maps with shadowing, the $\omega$-limit sets are precisely those sets having internal chain transitivity.

Détail

DOI : 10.4064/fm217-1-4

Keywords: continuous map compact metric space set subset internally chain transitive every every delta there sequence points langle ldots rangle i delta leq paper prove tent maps periodic critical point every closed internally chain transitive set necessarily omega limit set furthermore there least countably many tent maps non recurrent critical point which there closed internally chain transitive set which omega limit set together these results lead conjecture tent maps shadowing omega limit sets precisely those sets having internal chain transitivity