We introduce the concept of weakly mixing sets of order n and show that, in contrast to weak mixing of maps, a weakly mixing set of order n does not have to be weakly mixing of order n + 1. Strictly speaking, we construct a minimal invertible dynamical system which contains a non-trivial weakly mixing set of order 2, whereas it does not contain any non-trivial weakly mixing set of order 3. In dimension one this difference is not that much visible, since we prove that every continuous...

We study the preservation of the periodic orbits of an $A$-monotone tree map $f:T⟶T$ in the class of all tree maps $g:S⟶S$ having a cycle with the same pattern as $A$. We prove that there is a period-preserving injective map from the set of (almost all) periodic orbits of $f$ into the set of periodic orbits of each map in the class. Moreover, the relative positions of the corresponding orbits in the trees $T$ and $S$ (which need not be homeomorphic) are essentially preserved.

This paper is the second part of [2] and is devoted to the study of the spiral orbits of self maps of the 4-star with the branching point fixed, completing the characterization of the strongly directed primary orbits for such maps.

This paper is devoted to a systematic study of a class of binary trees encoding the structure of rational numbers both from arithmetic and dynamical point of view. The paper is divided into three parts. The first one is mainly expository and consists in a critical review of rather standard topics such as Stern-Brocot and Farey trees and their connections with continued fraction expansion and the question mark function. In the second part we introduce two classes of (invertible and non-invertible)...