A generalization of a theorem of Janos and Edelstein

Abstract:

A generalization of Edelstein’s version of a theorem of Janos and its converse are obtained: Theorem. Let X be a compact metrizable topological space, and f, g be continuous self-maps of X such that $gf = fg$ and f is bijective. Then g is injective and $\bigcap \nolimits _1^\infty {{g^n}X = \{ {x_0}\} }$, where ${x_0} \in X$, iff, given $\lambda ,0 < \lambda < 1$, a homeomorphism h of X into ${l_2}$ exists such that \[ \left \| {h(gx) - h(gy)} \right \| = \lambda \left \| {h(fx) - h(fy)} \right \|\] for all $x,y \in X$.