For C^0 generic continuous maps or homeomorphisms on compact Riemannian manifolds, physical-like measures coincide with invariant measures supported on chain recurrent classes.
Thompson, The irregular set for maps with the specification property has full topolog- ical pressure, Dyn
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Physical-like Measures Coincide with Invariant Measures Supported on Chain Recurrent Classes
For C^0 generic continuous maps or homeomorphisms on compact Riemannian manifolds, physical-like measures coincide with invariant measures supported on chain recurrent classes.