Chaos on the interval in terms of distributional functions

In paper [2], Balibrea and Smítal studied distributional functions of a continuous map f of the interval I and they proposed to use the “size" of the set of all distributional functions F (f) (with respect to different pairs of points) as a measure of chaos of the map f . We show that not the size of F (f) itself, but the existence of a distributional function which is strictly increasing, can be used to detect chaotic behavior of the interval map f . The main result shows that any topologically transitive unimodal map of the interval generates strictly increasing distributional function. On the other hand, the existence of a strictly increasing distributional function for f implies positive topological entropy of f , which proves the conjecture from [5] for interval maps.