Paradoxical functions on the interval

Abstract:

In this paper it is shown that any expanding with Lipschitz derivative function $f$ has a contradictory behaviour from the point of view of chaos in the sense of Li and Yorke. On the one hand it cannot generate scrambled sets of positive Lebesgue measure. On the other hand the two-dimensional set $\operatorname {Ch} (f)$ including the pairs $(x,y)$ such that $\{ x,y\}$ is a scrambled set of $f$ has positive measure. In fact, both the geometric structure (almost everywhere) and measure of $\operatorname {Ch} (f)$ can be explicitly obtained.