On the irregular points for systems with the shadowing property

• DOI: 10.1017/etds.2016.126
• Publication Date: 2017-03-14T12:32:42Z
• Authors: YIWEI DONG; PIOTR OPROCHA; XUETING TIAN
• Keywords: Lebesgue measure; Topological Entropy

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