AN ALMOST EVERYWHERE VERSION OF SMÍTAL’S ORDER–CHAOS DICHOTOMY FOR INTERVAL MAPS

Abstract

AbstractWe prove that iff:I=[0,1]→Iis aC3-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 measureλ; (ii) bothS(f) andScramb(f) have positive measure. HereR(f),S(f), andScramb(f) respectively stand for the set of approximately periodic points off, the set of sensitive points to the initial conditions off, and the two-dimensional set of points (x,y) such that {x,y} is a scrambled set forf. Also, we show that iffis piecewise monotone and has no wandering intervals, then eitherλ(R(f))=1 orλ(S(f))>0, and provide examples of mapsg,hof this type satisfyingS(g)=S(h)=Isuch that, on the one hand,λ(R(g))=0 andλ2(Scramb(g))=0 , and, on the other hand,λ(R(h))=1 .