On backward attractors of interval maps*

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