  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.
