Entropy of induced maps of regular curves homeomorphisms

Let f : X → be a self homeomorphism of continuum X, we show that the topological entropy induced system (2X, 2f) is infinite provided X\Ω(f) not empty. If furthermore regular curve then it shown has if and only Moreover prove for (C(X), C(f)) equivalence between following properties: (i) zero entropy; (ii) there no Li-Yorke pair (iii) any periodic subcontinnum A connected component C X\Ω(f), ⊂ ∩ ≠ ∅. In particular, either or two possible values 0 ∞. At end, give an example pointwise rational F Y with map C(F).