Factor maps and invariant distributional chaos

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.