  Given a vertex colouring of the infinite $n$-ary Cantor tree with $m$ colours
($n,m\geq 2$), the natural problem arises: may this colouring induce a
bijective colouring of the infinite paths starting at the root, i.e., that
every infinite $m$-coloured string is used for some of these paths but
different paths are not coloured identically? In other words, we ask if the
above vertex colouring may define a bijective short map between the
corresponding Cantor spaces. We show that the answer is positive if and only if
$n\geq m$, and provide an effective construction of the bijective colouring in
terms of Mealy automata and functions defined by such automata. We also show
that a finite Mealy automaton may define such a bijective colouring only in the
trivial case, i.e. $m=n$.
