A note on the variational principle for preimage pressures

In this paper, two types of variational principles for preimage pressure are considered for non-invertible systems. We first consider the variational principles from the convex analysis viewpoints and investigate the existence of the equilibrium state. Secondly, we obtain a variational principle of the preimage pressure for certain sub-additive potentials, and show that the corresponding preimage pressure coincides with that for a single bounded observable.