  Recently, Kiriki, Nakano and Soma introduced a concept called pointwise
emergence as a new quantitative perspective into the study of non-existence of
averages for dynamical systems. In the present paper, we consider the set of
points with high pointwise emergence for systems with non-uniform structure and
prove that this set carries full topological pressure. For the proof of this
result, we show that such systems have ergodic measures of arbitrary
intermediate pressures.
