Let ( X , τ ) (X,\tau ) be a metrizable topological space, P ( τ ) \mathcal {P}(\tau ) be the family of all metrics on X whose metric topologies are τ \tau . Assume that the semigroup F of maps from X into itself, with composition as its semigroup operation, is equicontinuous under some d ∈ P ( τ ) d \in \mathcal {P}(\tau ) ; then we have the following results: I. There exists d ′ ∈ P ( τ ) d’ \in \mathcal {P}(\tau ) such that f is nonexpansive under d ′ d’ for each f ∈ F f \in F . II. If F is countable, commutative, and for each f ∈ F f \in F , there is x f ∈ X {x_f} \in X such that the sequence ( f n ( x ) ) n = 1 ∞ ({f^n}(x))_{n = 1}^\infty converges to x f , ∀ x ∈ X {x_f},\forall x \in X , then there exists d ∈ P ( τ ) d \in \mathcal {P}(\tau ) such that f is contractive under d d for each f ∈ F f \in F . III. If there is p ∈ X p \in X such that (1) lim n → ∞ f n ( x ) = p , ∀ x ∈ X {\lim _{n \to \infty }}{f^n}(x) = p,\forall x \in X and ∀ f ∈ F \forall f \in F , (2) there is a neighbourhood B of p such that lim m → ∞ f n 1 f n 2 ⋯ f n m ( B ) = { p } {\lim _{m \to \infty }}{f_{{n_1}}}{f_{{n_2}}} \cdots {f_{{n_m}}}(B) = \{ p\} for any choice of f n i ∈ F , i = 1 , … , m {f_{{n_i}}} \in F,i = 1, \ldots ,m , and the limit depends on m only, then for each λ \lambda with 0 > λ > 1 0 > \lambda > 1 , there exists d ′ ∈ P ( τ ) d’ \in \mathcal {P}(\tau ) such that each f in F is a Banach contraction under d ′ d’ with Lipschitz constant λ \lambda .