On variants of distributional chaos in dimension one

In their famous paper from 1994, B. Schwaizer and J. Smital, [B. Schwaizer and J. Smital, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (1994), pp. 737-754] fully characterized topological entropy of interval maps in terms of distribution functions of distance between trajectories. Strictly speaking, they proved that a continuous map f :[0, 1]->[0, 1] has zero topological entropy if and only if for every x, y is an element of [0, 1] the following limit exists: lim(n -> infinity) 1/n vertical bar{0 <= i < n : d (f(i)(x),f(i)(y)) < t}vertical bar for every real number t except at most countable set. While many partial efforts have been made in previous years, still there is no proof that the result of Schwaizer and Smital holds on every topological graph. Here we offer the proof of this fact, filling a gap existing in the literature of the topic.