On ωω-limit sets of non-autonomous discrete systems on trees

Let TT be a tree and fnfn be a sequence of continuous maps from TT to TT which converges uniformly to a continuous map ff. In this note we show that if every periodic point of ff is a fixed point, then for every x∈Tx∈T, ω(x,fn)ω(x,fn), ωω-limit set of xx under (T,fn)(T,fn), is a closed connected subset of TT and every point of ω(x,fn)ω(x,fn) is a fixed point of ff.