On lipschitz extension of interval maps

Abstract

Let [a, b] be a compact real interval andf: [a, b]→[a, b] a continuous map from [a, b] into itself. We say thatf is topologically mixing if for any openU, V⊂[a, b] there exists anN such thatf ⁿ (U)∩V≠ϕ for anyn>N. Denote byA(f) the set of those from pointsa, b which have no preimages in (a, b) and ent(f) the topological entropy off. We show the following: Iff: [a, b]→[a, b] satisfies the conditions (i)f is topologically mixing, (ii)A(f)=ϕ, (iii) ent(f)=logν<∞, thenf has a ν-Lipschitz extension, i.e. there exist a ν-Lipschitz mapg: [a, b]→[a, b] and a nondecreasing surjective maph: [a, b]→[a, b] such thatf∘h=h∘g.