Stability of the distribution function for piecewise monotonic maps on the interval (Q1661077)

Distributional chaos and the upper and lower distribution functions for continuous interval maps were introduced by \textit{B. Schweizer} and \textit{J. Smítal} [Trans. Am. Math. Soc. 344, No. 2, 737--754 (1994; Zbl 0812.58062)]. The paper reviewed here considers piecewise monotone interval maps, possibly with discontinuities, and defines the approximating distribution function. The authors prove that the approximating distribution function of a mixing basic set agrees with the lower distribution function if the lower distribution function is right continuous. Furthermore, they prove that the approximating distribution function for mixing basic sets is upper semi-continuous. This implies the stability of distributional chaos (defined by the approximating distribution functions) under small perturbations of the map.