Ergodic average of typical orbits and typical functions

In this article we mainly aim to know what kind of asymptotic behavior typical orbits can display. For example, show in any transitive system, the empirical measures a orbit cover all dense and intersect some physical-like measures. particular, if union set is not singleton, then will display historic simultaneously for continuous functions limit ergodic average along every function equals closed interval composed by sets on orbits. Moreover, contains measures, above rotation set. These results are only suitable systems with specification-like properties or minimal systems, but also many other including general (not assumed uniformly hyperbolic) nontrivial homoclinic classes Bowen eyes. introduce new property called $ m $-$ g $-product weaker than classical specification property, examples constructed.