Free ergodic amenable actions with positive Rokhlin entropy have infinite complex L1- and L10-orbit multiplicity, so no finite family of functions has dense Koopman orbit span.
Iwanik,The problem ofL p-simple spectrum for ergodic group automorphisms, Bull
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Positive Rokhlin Entropy Implies Infinite $L^1$-Orbit Multiplicity: A Negative Answer to Thouvenot's Question
Free ergodic amenable actions with positive Rokhlin entropy have infinite complex L1- and L10-orbit multiplicity, so no finite family of functions has dense Koopman orbit span.