DECOMPOSING ISOMETRIC EXTENSIONS USING GROUP EXTENSIONS

This paper studies the structure of isometric extensions of compact metric topological dynamical systems with Z action and gives two decompositions of the general case to a more structured case. Suppose that Y --> X is a M-isometric extension. An extension, Z, of Y is constructed which is also a G-isometric extension of X, where G is the group of isometries of M. The first construction shows that, provided that (X, T) is transitive, there are almost-automorphic extensions Y' --> Y and X' --> X, so that Y' is homeomorphic to X' x M and the natural projection Y' --> X' is a group extension. The second shows that, provided that (X, T) is minimal, there is a G-action on Z which commutes with T and which preserves fibres and acts on each of them minimally. Each individual orbit closure, Z(a), in Z is a G'-isometric extension of X, where G' is a subgroup of G, and there is a G'-action on Z(a) which commutes with T, preserves fibres and acts minimally on each of them. Two illustrations are presented. Of the first: to reprove a result of Furstenberg; that every distal point is IP*-recurrent. Of the second: to describe the minimal subsets in isometric extensions of minimal topological dynamical systems.