On notions of determinism in topological dynamics

TL;DR

This paper explores the relationship between topological entropy, determinism, and prediction in topological dynamics, revealing that zero entropy is the key restriction for invariant measures and extending the theory to multidimensional systems.

Contribution

It introduces a new method linking topological determinism and zero entropy, and extends the theory to multidimensional systems and symbolic dynamics.

Findings

Topological determinism imposes no restriction on invariant measures except zero entropy.

Unique future prediction only occurs in finite systems.

Constructed a zero-entropy system with a globally supported ergodic measure where points have multiple preimages.

Abstract

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.