On backward attractors of interval maps

Special α -limit sets ( sα -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 sα -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 centre that completely characterizes which interval maps have all sα -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 ω -limit sets to solve four additional conjectures by Kolyada, Misiurewicz, and Snoha relating topological properties of sα -limit sets to the dynamics within them. For example, we show that the isolated points in a sα -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 sα -limit set is nowhere dense. Moreover, we show that sα -limit sets in the interval are always both F σ and G δ . Finally, since sα -limit sets need not be closed, we propose a new notion of β -limit sets to serve as backward attractors. The β -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 sα -limit set. At the end of the paper we suggest several new problems about backward attractors.