Variational principle of random pressure function

Abstract: This paper aims to develop a convex analysis approach to random pressure functions of random dynamical systems. Using some convex analysis techniques and functional analysis, we establish a variational principle for random pressure function, which extends Bis et al. work (A convex analysis approach to entropy functions, variational principles and equilibrium states, Comm. Math. Phys. (2022) \textbf{394} 215-256) and Ruelle's work (Statistical mechanics on a compact set with $\mathbb{Z}^{\nu}$ action satisfying expansiveness and specification, Trans. Amer. Math. Soc. (1973) \textbf{187} 237-251) to random dynamical systems. The present paper provides a strategy of obtaining some proper variational principles for entropy-like quantities of dynamical systems to link the topological dynamics and ergodic theory. As applications, we establish variational principles of maximal pattern entropy and polynomial topological entropy of zero entropy systems of $\mathbb{Z}$-actions, mean dimensions of infinite entropy systems acting by amenable groups and preimage entropy-like quantities of non-convertible random dynamical systems.