Factor maps and invariant distributional chaos

TL;DR

This paper demonstrates that maps with the specification property possess invariant distributionally scrambled sets and that these sets can be transferred via finite-to-one factor maps, with applications to certain differential equations.

Contribution

It establishes the existence and transferability of invariant distributionally scrambled sets for maps with the specification property, addressing open questions in the field.

Findings

Maps with specification property have invariant distributionally scrambled sets.

Such scrambled sets can be transferred through finite-to-one factor maps.

Application demonstrated on Poincaré maps of specific differential equations.

Abstract

The main aim of this article is to show that maps with specification property have invariant distributionally scrambled sets and that this kind of scrambled set can be transferred from factor to extension under finite-to-one factor maps. This solves some open questions in the literature of the topic. We also show how our method can be applied in practice, taking as example Poincar\'{e} map of time-periodic nonautonomous planar differential equation $$ \dot{z}=v(t,z)=(1+e^{i\kappa t}\abs{z}^2)\z^3 -N $$ where $\kappa$ and $N$ are sufficiently small positive real numbers.