Preprint MA 67/2009

On variants of distributional chaos in dimension one

Michal Málek, Piotr Oprocha

In their famous paper from 1994, B. Schwaizer and J. Smítal 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 ∈ [0,1] the following limit exists

limn → ∞ 1/n # {0 ≤ i < n : d(f i(x), f i(y)) < t }

for every real number t except at most countable set. While many steps have been made in previous years, still there is no proof that the result of Schwaizer and Smítal holds on every topological graph. Here we offer the proof of this fact, filling a gap existing in the literature of the topic.