\(\omega\)-limit sets, nonwandering sets of tree maps (Q2706883)

Let \(T\) be a tree and \(f\) be a continuous map \(T\to T\). In the dynamical system \((T,f)\), \(\Omega\) the set of non-wandering points, \(\Lambda\) the union of \(\{\omega(x) :x\in X\}\), \(\Pi\) the union of \(\{\gamma(x) :x\in X\}\) where \(\gamma(x)= \omega(x) \cap\alpha (x)\).NEWLINENEWLINENEWLINETheorem: 1. \(\Omega-\Pi\) is countable. 2. \(\Lambda-\Gamma\), \(\overline P-P\) are either empty or countably infinite where \(P\) is the set of periodic points in \((T,f)\).