On the irregular points for systems with the shadowing property

TL;DR

This paper demonstrates that for systems with the shadowing property, irregular points with divergent Birkhoff averages have full entropy, but are non-observable and carry infinite entropy, contrasting with observable points.

Contribution

It establishes the full entropy of irregular points in systems with shadowing and characterizes their measure-theoretic properties in generic maps.

Findings

Irregular points have full entropy in shadowing systems.

Observable points with convergent averages are typical in generic maps.

Irregular points are non-observable but possess infinite entropy.

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.