For amenable group actions, IE-tuples of the induced measure system equal the closed convex hull of IE-tuples of the original system, with partial analogues for IN and IT tuples and a sharp n ≤ C log m embedding bound for ℓ_q^n inside ℓ_∞^m.
Topological mean dimension of induced systems
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For a topological system with positive topological entropy, we show that the induced transformation on the set of probability measures endowed with the weak-$*$ topology has infinite topological mean dimension. We also estimate the rate of divergence of the entropy with respect to the Wasserstein distance when the scale goes to zero.
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Local entropy theory, combinatorics, and local theory of Banach spaces
For amenable group actions, IE-tuples of the induced measure system equal the closed convex hull of IE-tuples of the original system, with partial analogues for IN and IT tuples and a sharp n ≤ C log m embedding bound for ℓ_q^n inside ℓ_∞^m.