On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps

Let f:I=[0,1]→ I be a Borel measurable map and letμ be a probability measure on the Borel subsets of I. Weconsider three standard ways to cope with the idea of ``observablechaos'' for f with respect to the measure μ: h_μ(f)>0---when μ is invariant---, μ(L⁺(f))>0---when μ is absolutely continuous with respect to the Lebesguemeasure---, and μ(S^μ(f))>0. Here h_μ(f), L⁺(f) andS^μ(f) denote, respectively, the metric entropy of f, theset of points with positive Lyapunov exponent, and the set ofsensitive points to initial conditions with respect to μ.    It is well known that if h_μ(f)>0 or μ(L⁺(f))>0, thenμ(S^μ(f))>0, and that (when μ is invariant andabsolutely continuous) h_μ(f)>0 and μ(L⁺(f))>0 areequivalent properties. However, the available proofs in theliterature require substantially stronger hypotheses than thosestrictly necessary. In this paper we revisit these notions andshow that the above-mentioned results remain true in, essentially,the most general (reasonable) settings. In particular, we improvesome previous results from [2], [6], and[23].