Has this metric been considered anywhere?

Question

Let $X$ be a compact metric space and denote by $d$ the metric on $X$. I wondered whether the following metric $d_\infty : C(X,X)\times C(X,X) \rightarrow \mathbb{R_0^+}$ given by $$d_\infty (f,g)= \sup_{x\in X,k\in\mathbb{N}}d(f^k(x),g^k(x))$$ has been considered and if it has a name in the literature. Thanks