Physical-like Measures Coincide with Invariant Measures Supported on Chain Recurrent Classes

Abstract: For $C^0$ generic continuous maps or homeomorphisms on compact Riemannian manifold, we prove that (1) the space of physical-like measures coincides with the set of invariant measures supported on chain recurrent classes, (2) every point in the base space is typical (that is, for any point in the base space, its empirical measures are contained in the space of physical-like measures) and (3) there is a subset of strongly regular set with Lebesgue zero measure but has infinite topological entropy. Moreover, some comparison between $C^0$ generic systems and $C^0$ conservative generic systems are discussed.