On Notions of Determinism in Topological Dynamics

We examine the relation between topological entropy, invertability, and prediction in topological dynamics. We show that topological determinism in the sense of Kamińsky Siemaszko and Szymański 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.