We consider which spaces can be realized as the omega limit set of the discrete time dynamical system. This is equivalent to asking which spaces admit a chain transitive homeomorphism and which do not. This leads us to ask for spaces where all homeomorphisms are chain transitive.
A remark on omega limit sets for non-expansive dynamics
The invariant random order extension property is equivalent to amenability
Additive, almost additive and asymptotically additive potential sequences are equivalent
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