On the irregular points for systems with the shadowing property

Abstract: We prove that when $f$ is a continuous selfmap acting on compact metric space $(X,d)$ which satisfies the shadowing property, then the set of irregular points (i.e. points with divergent Birkhoff averages) has full entropy. Using this fact we prove that in the class of $C^0$-generic maps on manifolds, we can only observe (in the sense of Lebesgue measure) points with convergant Birkhoff averages. In particular, the time average of atomic measures along orbit of such points converges to some SRB-like measure in the weak$^*$ topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.