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, Irregular sets, the β-transformation and the almost specification property , Transactions of the American Mathematical Society, 2012, 3 64 (10): 5395-5414
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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.