Stability of the distribution function for piecewise monotonic maps on the interval

Michal Málek,Peter Raith

doi:10.3934/dcds.2018105

Abstract

For piecewise monotonic maps the notion of approximating distribution function is introduced. It is shown that for a mixing basic set it coincides with the usual distribution function. Moreover, it is proved that the approximating distribution function is upper semi-continuous under small perturbations of the map.

Highlights

The notion of distributional chaos has been introduced by Schweizer and Smıtal in [12]
Given t ∈ R and two points x, y in a dynamical system one counts the relative number of times when the distance of the orbits of x and y is smaller than t
More details on distributional chaos can be found in the nice survey paper [11] and in [3]

Summary

Introduction

The notion of distributional chaos has been introduced by Schweizer and Smıtal in [12]. For a maximal irreducible C ⊆ D let K(C) be the set of all x such that x is represented by an infinite path in C and x is not contained in the interior of an interval I satisfying that T n I is monotonic for all n. Let Dn be the smallest set containing Y and with the property that D ∈ Dn and D → C imply C ∈ Dn. Let (D∞, →) be the Markov diagram of T X∞ with respect to Ym. Since B ⊆ X∞ is mixing there exists a maximal irreducible and aperiodic C ⊆ D∞ such that B = K(C).

Since both

By the compactness of there exists a k

Using the tk for r this implies

Yn with x

Therefore we obtain