Eventually Shadowable Points, Orbital Shadowable Points and Entropy Points

In this article, we explore several sufficient conditions under which an eventually shadowable point can be approximated by entropy points. Moreover, we derive distinct conditions for an eventually shadowable point to be an entropy point. We also study the situation where a system related to an eventually shadowable point is conjugated to an odometer. On the other hand, we introduce the concept of orbital shadowable points and obtain the following results: The set of orbital shadowable points of a homeomorphism on a compact metric space is invariant; a homeomorphism has the orbital shadowing property if and only if every point is orbital shadowable; a homeomorphism has the orbital shadowing property if and only if the set of orbital shadowable points has full measure with respect to every ergodic measure.