On the irregular points for systems with the shadowing property

Abstract

We prove that when $f$ is a continuous self-map acting on a compact metric space $(X,d)$ that 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 convergent Birkhoff averages. In particular, the time average of atomic measures along orbits of such points converges to some Sinai–Ruelle–Bowen-like measure in the weak$^{\ast }$ topology. Moreover, such points carry zero entropy. In contrast, irregular points are non-observable but carry infinite entropy.