On a certain map of a triangle

Volume 155 / 1998

Grzegorz Świrszcz Fundamenta Mathematicae 155 (1998), 45-57 DOI: 10.4064/fm-155-1-45-57


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 $∪F^{-n}(I)$ of Δ and there is another dense set Λ consisting of points whose orbits approach the interval I but are not attracted by I.


  • Grzegorz Świrszcz

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