arXiv subjects
Publications and source records attributed to Sina Greenwood.
A CR-dynamical system is a pair $(X, G)$, where $X$ is a compact metric space and $G$ is a closed relation (CR) on $X$. In this paper, we introduce a new type of transitive point and transitivity in CR-dynamical systems. We develop a new tool called transitivity trees, which we use to determine the relationship between the different types of transitive points.
Let $X$ be a Lelek fan or a Cantor fan and let $Y$ be a Lelek fan or a Cantor fan. In this paper, we study embeddings $ f: X \to Y $ that admit retractions from $ Y $ onto $ f(X)$. In 1989, W. J. Charatonik and J. J. Charatonik proved that if $X$ is a Lelek fan and $Y$ is a Cantor fan, then no embedding $f$ of $X$ into $Y$ admits a retraction from $Y$ onto $f(X)$. They also showed that if both $X$ and $Y$ are Cantor fans, then every embedding $f$ of $X$ into $Y$ admits such a retraction. In this paper, we address the two remaining cases. First, we consider the situation where $X$ is a Cantor fan and $Y$ is a Lelek fan. We prove that in this case, every embedding $f$ of $X$ into $Y$ admits a retraction from $Y$ onto $f(X)$. Second, we examine the case where both $X$ and $Y$ are Lelek fans. Here, we show that there exist embeddings $f$ that do admit a retraction from $Y$ onto $f(X)$, as well as embeddings that do not. For this latter case, we also identify additional properties of embeddings that ensure the existence of a retraction from $Y$ onto $f(X)$.
We study different types of transitive points in CR-dynamical systems (X,G) with closed relations G on compact metric spaces X. We also introduce transitive and dense orbit transitive CR-dynamical systems and discuss their properties and the relations between them. This generalizes the notion of transitive topological dynamical systems (X, f ).
For a continuous function $f : [0,1] \to [0,1]$ we define a splitting sequence admitted by $f$ and show that the inverse limit of $f$ is an arc if and only if $f$ does not admit a splitting sequence.