The work of Lewis Bowen on the entropy theory of non-amenable group actions

Abstract

We present the achievements of Lewis Bowen, or, more precisely, his breakthrough works after which a theory started to develop. The focus will therefore be made here on the isomorphism problem for Bernoulli actions of countable non-amenable groups which he solved brilliantly in two remarkable papers. Here two invariants were introduced, which led to many developments. DEFINITION 1.2. We say that two actions (X , A , µ,G, τ). and (Y , B, ν,G, σ) of the same group G are isomorphic if there exists an invertible measure preserving mapping Φ from (X , A , µ) to (Y , B, ν) such that σ g Φ = Φτ g for almost every x ∈ X , for all g ∈ G. In case the mapping Φ is not invertible, we say that (Y , B, ν,G, σ) is a factor of (X , A , µ,G, τ).

Entropy invariants of generic actions

“…Among the wide variety of the entropy notions (see, for example, [4], [11], [22], [24]) we consider slightly modified invariants of Kushnirenko [13].…”