On the $\omega$-limit sets of tent maps
Volume 217 / 2012
Abstract
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.
