On distributional spectrum of piecewise monotonic maps

TL;DR

This paper investigates a class of piecewise monotonic interval maps, demonstrating that they always exhibit distributional chaos and have finite distributional and weak spectra, extending known results from continuous maps.

Contribution

It introduces a broad class of piecewise monotonic maps with specific properties and proves that their distributional spectra are finite, generalizing previous results for continuous maps.

Findings

Distributional chaos is always present in the class.

Distributional and weak spectra are finite for these maps.

Examples show the conditions cannot be weakened.

Abstract

We study a certain class of piecewise monotonic maps of interval. These maps are strictly monotone on finite interval partition, satisfies 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. Main result states that distributional spectrum as well as weak spectrum is always finite. This is a generalization of same result for continuous maps on the interval, circle and tree. Examples showing that conditions on mentions class can not be weakened are presented.