|dc.description.abstract||The classification of minimal sets is a central theme in
abstract topological dynamics. Recently this work has been
strengthened and extended by consideration of homomorphisms.
Background material is presented in Chapter I. Given a
flow on a compact Hausdorff space, the action extends naturally
to the space of closed subsets, taken with the Hausdorff
topology. These hyperspaces are discussed and used to give a
new characterization of almost periodic homomorphisms.
Regular minimal sets may be described as minimal subsets
of enveloping semigroups. Regular homomorphisms are defined
in Chapter II by extending this notion to homomorphisms with
minimal range. Several characterizations are obtained.
In Chapter III, some additional results on homomorphisms
are obtained by relativizing enveloping semigroup notions.
In Veech's paper on point distal flows, hyperspaces are
used to associate an almost one-to-one homomorphism with a
given homomorphism of metric minimal sets. In Chapter IV, a
non-metric generalization of this construction is studied in
detail using the new notion of a highly proximal homomorphism.
An abstract characterization is obtained, involving only the
abstract properties of homomorphisms. A strengthened version
of the Veech Structure Theorem for point distal flows is
In Chapter V, the work in the earlier chapters is
applied to the study of homomorphisms for which the almost
periodic elements of the associated hyperspace are all
finite. In the metric case, this is equivalent to having
at least one fiber finite. Strong results are obtained by
first assuming regularity, and then assuming that the relative
proximal relation is closed as well.||en_US