In their recent paper [KNS2019], the first author, S. Kiriki, and T. Soma
introduced a concept of pointwise emergence to measure the complexity of
irregular orbits. They constructed a residual subset of the full shift with
high pointwise emergence. In this paper we consider the set of points with high
pointwise emergence for topologically mixing subshifts of finite type. We show
that this set has full topological entropy, full Hausdorff dimension, and full
topological pressure for any Hölder continuous potential. Furthermore, we
show that this set belongs to a certain class of sets with large intersection
property. This is a natural generalization of [FP2011] to pointwise emergence
and Carathéodory dimension.