Intra-orbit separation of dense orbits of dendrite maps

In this note, we will generalize the notion of separation index, which was introduced by Manjunath et al. (2006), to dendrite maps, and use the notion to characterize the intra-orbit separation for the orbits of continuous transitive dendrite maps. We will show: (i) For a dendrite map f, the separation index γ is greater than zero if and only if the set of fixed points of f is not an arc connected subspace. (ii) If the separation index γ of a dendrite map f is greater than zero, then for every 0<τ<γ and any pair of distinct points x and y on a dense orbit, {x,y} is a Li–Yorke pair of modulus τ.