  We examine the relation between topological entropy, invertability, and
prediction in topological dynamics. We show that topological determinism in the
sense of Kamisky Siemaszko and Szymaski imposes no restriction on invariant
measures except zero entropy. Also, we develop a new method for relating
topological determinism and zero entropy, and apply it to obtain a
multidimensional analog of this theory. We examine prediction in symbolic
dynamics and show that while the condition that each past admit a unique future
only occurs in finite systems, the condition that each past have a bounded
number of future imposes no restriction on invariant measures except zero
entropy. Finally, we give a negative answer to a question of Eli Glasner by
constructing a zero-entropy system with a globally supported ergodic measure in
which every point has multiple preimages.
