Predictability, topological entropy and invariant random orders

TL;DR

This paper establishes that topologically predictable actions of amenable groups have zero entropy, explores invariant random orders, and extends entropy formulas, contributing to the understanding of dynamical systems and group actions.

Contribution

It proves a conjecture linking predictability and zero entropy for amenable groups and develops a unified entropy formula involving invariant random orders.

Findings

Predictable amenable actions have zero topological entropy

Invariant random orders are connected to entropy formulas

Topologically prime sofic group actions have non-positive entropy

Abstract

We prove that a topologically predictable action of a countable amenable group has zero topological entropy, as conjectured by Hochman. On route, we investigate invariant random orders and formulate a unified Kieffer-Pinsker formula for the Kolmogorov-Sinai entropy of measure preserving actions of amenable groups. We also present a proof due to Weiss for the fact that topologically prime actions of sofic groups have non-positive topological sofic entropy.