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

Source abstract

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 .