Symbol sequences play a prominent role in the context of symbolic dynamics. Important features of a dynamical system are reflected by related statistics of subsequences. A dynamical behavior giving rise to a self-similar attractor and universal scaling relations, expressed by critical exponents, will lead to self-similar statistics of subsequences. In the present paper we show how self-similar distributions of subsequences, i.e., temporal self-similarity, can be connected with a scaling relation for dynamical entropies. Moreover, the effect of slightly perturbing perfectly self-similar sequences by contaminating them with noise is investigated. The achieved results are of importance for physical processes marking the borderline between order and chaos. \textcopyright{} 1996 The American Physical Society.