An Almost Everywhere Version of Smítal’s Order–chaos Dichotomy for Interval Maps

Abstract

We prove that if f : I = [0, 1] → I is a C 3 -map with negative Schwarzian derivative, nonflat critical points and without wild attractors, then exactly one of the following alternatives must occur: (i) R( f ) has full Lebesgue measure λ; (ii) both S( f ) and Scramb( f ) have positive measure. Here R( f ), S( f ), and Scramb( f ) respectively stand for the set of approximately periodic points of f , the set of sensitive points to the initial conditions of f , and the two-dimensional set of points (x, y) such that {x, y} is a scrambled set for f . Also, we show that if f is piecewise monotone and has no wandering intervals, then either λ(R( f )) = 1 or λ(S( f )) > 0, and provide examples of maps g, h of this type satisfying S(g) = S(h) = I such that, on the one hand, λ(R(g)) = 0 and λ 2 (Scramb(g)) = 0, and, on the other hand, λ(R(h)) = 1.2000 Mathematics subject classification: primary 37E05; secondary 37C05, 37D45, 37E15.