Intra-orbit separation of dense orbits of interval maps

Abstract

We introduce the notion of intra-orbit separation for the orbits of continuous transitive maps on a compact interval to demonstrate separation of two points on a given dense orbit. We associate a non-negative real number γ with a transitive interval map ƒ called the separation index of the map ƒ. For a transitive map ƒ having at least two fixed points we show: (i) the separation index γ is positive, (ii) for every 0 < τ < γ and any pair of distinct points x_j and x_l on a dense orbit, (x_j, x_l) is a Li-Yorke pair of modulus τ, that is, lim sup_n→+∞|ƒⁿ(x) - ƒⁿ(y)| > τ and lim inf_n→+∞|ƒⁿ(x) − ƒⁿ(y)| = 0.