Endomorphisms of unital C*-algebras may be viewed via Gelfand-Naimark theorem as natural generalizations of continuous maps of compact spaces. We will recall this interpretation and then discuss evolutions on a particular class of C*-algebras, the so-called Cuntz algebras. The corresponding endomorphisms turn out to be strongly related to continuous transformations of the Cantor set (and to symbolic dynamics). We will present two perspectives on such connections: one related to the noncommutative entropy computations, and another to the classification of automorphisms of Cuntz algebras.