The critical exponent of a word is an important combinatorial concept which has applications in symbolic dynamics and transcendental number theory. It is natural to define a countable class of interval self-maps sending every real in [0,1] to the inverse of the critical exponent of its base-n expansion. In the investigation of these maps, an interesting interplay emerges between recent results on the dynamics of densely discontinuous maps and combinatorial properties of words. Moreover, these maps provide examples of semi-continuous maps with very rich dynamics. This can be a useful starting point for the systematic investigation of the topological dynamical properties of this kind of maps, which is yet to begin.