Negative limit sets of graph maps

Abstract

Let $G$ be a graph and $f:G\rightarrow G$ be a continuous map. A sequence $\theta=(x_0,x_{-1},\ldots,x_{-n},\ldots)$ of points in $G$ is called a negative orbit through $x_0$ of $f$ if $f(x_{-n})=x_{-n+1}$ for every positive integer $n$. The $\alpha$-limit set of $\theta$ is the set $\alpha(\theta,f)$ of all limit points of $\theta$. In this paper, we show that the $\alpha$-limit set of every negative orbit of $f$ is an $\omega$-limit set of some point in $G$. Besides, we illustrate that the above result does not hold for a dendrite map.