On iterates of fully closed absolutes

In this article we give a sufficient condition for a space X to have the fully closed absolute faX with the property f a ( f a X ) = f a X . An example of a compact space X such that the canonical mapping f a ( α + 1 ) X → f a ( α ) X (where α is a given ordinal) is not a homeomorphism is constructed. Also we give an example of a compact space X such that the canonical mapping f a X → X is not a homeomorphism but for which there exists a homeomorphism f a X → X .