On the bijective colouring of Cantor trees based on transducers

Given a vertex colouring of the infinite n-ary Cantor tree with m colours (n,m≥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≥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.