On distributional spectrum of piecewise monotonic maps

Abstract

We study a certain class of piecewise monotonic maps of an interval. These maps are strictly monotone on finite interval partitions, satisfy the Markov condition, and have generator property. We show that for a function from this class distributional chaos is always present and we study its basic properties. The main result states that the distributional spectrum, as well as the weak spectrum, is always finite. This is a generalization of a similar result for continuous maps on the interval, circle, and tree. An example is given showing that conditions on the mentioned class can not be weakened.