On Backward Attractors Of Interval Maps

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 centre 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.