This paper is indented to explain a dynamics on homotopies on the compact metric space, by the envelopes of homotopies. It generalizes the notion of not only the envelopes of maps in discrete geometry ([3]), but the envelopes of flows in continuous geometry ([5]). Certain distinctions among the homotopy geometry, the ow geometry and the discrete geometry will be illustrated. In particular, it is shown that any ${omega}$-limit set, as well as any attractor, for an envelope of homotopies is an empty set (provided the homotopies that we treat are not trivial), whereas it is nonempty in general in discrete case.