Abstract
Sofic entropy is an invariant for probability-preserving actions of sofic groups. It was introduced a few years ago by Lewis Bowen, and shown to extend the classical Kolmogorov–Sinai entropy from the setting of amenable groups. Some parts of Kolmogorov–Sinai entropy theory generalize to sofic entropy, but in other respects this new invariant behaves less regularly. This paper explores conditions under which sofic entropy is additive for Cartesian products of systems. It is always subadditive, but the reverse inequality can fail. We define a new entropy notion in terms of probability distributions on the spaces of good models of an action. Using this, we prove a general lower bound for the sofic entropy of a Cartesian product in terms of separate quantities for the two factor systems involved. We also prove that this lower bound is optimal in a certain sense, and use it to derive some sufficient conditions for the strict additivity of sofic entropy itself. Various other properties of this new entropy notion are also developed.
Highlights
Let G be a discrete sofic group, (X, μ) a standard probability space and T : G X a measurable action which preserves μ
This paper considers how hΣ behaves under forming Cartesian products of systems
The reverse implication is simple, so we focus on the forward implication
Summary
Let G be a discrete sofic group, (X, μ) a standard probability space and T : G X a measurable action which preserves μ. A closely related notion of convergence for a sequence of measures on model spaces already appears in [6] Using this notion, that paper gives a new formula for sofic entropy for certain special examples of probability-preserving systems and sofic approximations. We call hqΣ (μ) the model-measure sofic entropy of μ rel Σ Like sofic entropy, it is an isomorphism-invariant of the G-process (Theorem 6.4), and so it does not depend on the choice of the generating metric d. It is fairly easy to prove that hpΣs hdΣq, so most of the work goes into proving the reverse inequality This will be done by showing that individual good models for large Cartesian powers (μ×k, T ×k) can be converted into measures that doubly quenched-converge to μ. This is why we include this new formalism, rather than just using the definitions from [16], for example
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