On one of Birkhoff's theorems for backward limit points

In 1927 George Birkhoff in his book Dynamical Systems presented a theorem that describes the behaviour of trajectories outside of a set of non-wandering points on an arbitrary compacta. Much later in 1960s Sharkovsky followed up on Birkhoff’s work and published even stronger result, this time focusing on the set of omega limit points for interval maps. In this article we formulate similar statement for a neighbourhood of a set of different types of backward limit points for maps of the interval.