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 f called the separation index of the map f. For a transitive map f having at least two fixed points we show: (i) the separation index γ is positive, (ii) for every 0 τ and lim infn→+∞|fn(x) − fn(y)| = 0.