**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

lim_{n → ∞} ^{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.