Birkhoff centre and backward limit points - Rýžová, Veronika (Topology and its applications, 2023)
We suggest one complete and one partial solution to the selected problems presented in the recently published article On Backward Attractors of Interval Maps (Hantáková and Roth (2021) [15]). Specifically we prove a conjecture proposing a characterisation of sets of β-limit points (i.e. limit points of all accumulation points of backward orbit branches of a specific point) for graph maps. We show that β-limit sets coincide with Birkhoff centre Rec(f)‾ and that the condition for a point to belong to its β-limit set is equivalent to belonging to the β-limit set of an other point. In the second part of the paper we deal with genericity of having all sα-limit sets closed and we prove that maps with not all sα-limit sets closed are dense in C0([0,1]), which partially solves an open problem also suggested in the aforementioned article.