Naive entropy of dynamical systems

TL;DR

This paper investigates naive entropy as an invariant of dynamical systems, proving a conjecture relating naive entropy to sofic entropy for actions of sofic groups, and exploring properties of naive entropy in various systems.

Contribution

It proves the topological version of Bowen's conjecture linking naive entropy to sofic entropy for sofic group actions and analyzes naive entropy in different dynamical contexts.

Findings

Zero naive entropy implies sofic entropy at most zero for sofic group actions.

Generic free group actions on the Cantor set have sofic entropy at most zero.

Distal systems with an element of infinite order have zero naive entropy.

Abstract

We study an invariant of dynamical systems called naive entropy, which is defined for both measurable and topological actions of any countable group. We focus on nonamenable groups, in which case the invariant is two-valued, with every system having naive entropy either zero or infinity. Bowen has conjectured that when the acting group is sofic, zero naive entropy implies sofic entropy at most zero for both types of systems. We prove the topological version of this conjecture by showing that for every action of a sofic group by homeomorphisms of a compact metric space, zero naive entropy implies sofic entropy at most zero. This result and the simple definition of naive entropy allow us to show that the generic action of a free group on the Cantor set has sofic entropy at most zero. We observe that a distal $\Gamma$-system has zero naive entropy in both senses, if $\Gamma$ has an element of infinite order. We also show that the naive entropy of a topological system is greater than or equal to the naive measure entropy of the same system with respect to any invariant measure.