Emergence via non-existence of averages

Inspired by a recent work Berger, we introduce the concept of pointwise emergence. This provides with new quantitative perspective into study non-existence averages for dynamical systems. We show that high emergence on large set appears abundant systems: Any continuous maps compact metric space specification property have super-polynomial residual subset state space. Furthermore, there is dense any Newhouse open each element which has positive Lebesgue measure