Large intersection classes for pointwise emergence

In the paper [KNS2019], a concept of pointwise emergence was introduced to quantitatively study non-existence of averages, and a residual subset of the full shift with high pointwise emergence was constructed. 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 Holder 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 Caratheodory dimension.