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

## TL;DR

This paper proves that for generic continuous maps on compact manifolds, physical-like measures match invariant measures on chain recurrent classes, with implications for typical points and entropy, and compares conservative and non-conservative systems.

## Contribution

It establishes the equivalence of physical-like and invariant measures on chain recurrent classes for generic systems and explores their properties and differences.

## Key findings

- Physical-like measures coincide with invariant measures on chain recurrent classes.
- Every point's empirical measures are contained in the physical-like measure space.
- Existence of a zero Lebesgue measure set with infinite topological entropy.

## 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.