On backward attractors of interval maps

Abstract

Special $\alpha$-limit sets ($s\alpha$-limit sets) combine together all accumulation points of all backward orbit branches of a point $x$ under a noninvertible map. The most important question about them is whether or not they are closed. We challenge the notion of $s\alpha$-limit sets as backward attractors for interval maps by showing that they need not be closed. This disproves a conjecture by Kolyada, Misiurewicz, and Snoha. We give a criterion in terms of Xiong's attracting center that completely characterizes which interval maps have all $s\alpha$-limit sets closed, and we show that our criterion is satisfied in the piecewise monotone case. We apply Blokh's models of solenoidal and basic $\omega$-limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of $s\alpha$-limit sets to the dynamics within them. For example, we show that the isolated points in a $s\alpha$-limit set of an interval map are always periodic, the non-degenerate components are the union of one or two transitive cycles of intervals, and the rest of the $s\alpha$-limit set is nowhere dense. Moreover, we show that $s\alpha$-limit sets in the interval are always both $F_\sigma$ and $G_\delta$. Finally, since $s\alpha$-limit sets need not be closed, we propose a new notion of $\beta$-limit sets to serve as backward attractors. The $\beta$-limit set of $x$ is the smallest closed set to which all backward orbit branches of $x$ converge, and it coincides with the closure of the $s\alpha$-limit set. At the end of the paper we suggest several new problems about backward attractors.