  The first part of this work is devoted to the study of higher differentials
of pressure functions of H\"older potentials on shift spaces of finite type. By
describing the differentials of pressure functions via the Central Limit
Theorem for the associated random processes, we discover some rigid
relationships between differentials of various orders. The rigidity imposes
obstructions on fitting candidate convex analytic functions by pressure
functions of H\"older potentials globally, which answers a question of
Kucherenko-Quas. In the second part of the work we consider fitting candidate
analytic germs by pressure functions of locally constant potentials. We prove
that all 1-level candidate germs can be realised by pressures of some locally
constant potentials, as long as number of the symbolic set is large enough.
There are also some results on fitting 2-level germs by pressures of locally
constant potentials obtained in the work.
