Contractification of a semigroup of maps

Abstract:

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