On a certain map of a triangle

The paper answers some questions asked by Sharkovski concerning the map F:(u,v) ↦ (u(4-u-v),uv) of the triangle Δ = {u,v ≥ 0: u+v ≤ 4}. We construct an absolutely continuous σ-finite invariant measure for F. We also prove the following strange phenomenon. The preimages of side I = Δ ∩ {v=0} form a dense subset ${\displaystyle \cup F^{-n}\left(I\right)}$ of Δ and there is another dense set Λ consisting of points whose orbits approach the interval I but are not attracted by I.