arXiv subjects
Publications and source records attributed to Magdalena Foryś.
We show that for a non-trivial transitive dynamical system, it has a dense Mycielski invariant strongly scrambled set if and only if it has a fixed point, and it has a dense Mycielski invariant $δ$-scrambled set for some $δ>0$ if and only if it has a fixed point and not uniformly rigid. We also provide two methods for the construction of completely scrambled systems which are weakly mixing, proximal and uniformly rigid.
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 Poincaré map of time-periodic nonautonomous planar differential equation $$ \dot{z}=v(t,z)=(1+e^{iκt}\abs{z}^2)\z^3 -N $$ where $κ$ and $N$ are sufficiently small positive real numbers.