Variational principle for random pressure function

TL;DR

This paper introduces a variational principle for random pressure functions in dynamical systems, linking ergodic theory and topological dynamics through convex analysis techniques.

Contribution

It establishes a new variational principle for random pressure functions, connecting various entropy and dimension concepts in dynamical systems.

Findings

Established a variational principle for random pressure functions.

Linked ergodic theory with topological dynamics via this principle.

Derived variational principles for polynomial entropy, mean dimensions, and preimage entropy.

Abstract

For random dynamical systems, by summarizing the fundamental properties of Kifer's topological pressure we introduce the concept of random pressure functions, and define Ruelle's metric entropy for invariant measures. Employing the techniques from convex analysis and ergodic theory, we establish a variational principle for random pressure functions. Consequently, this new variational principle allows us to establish a vital bridge between ergodic theory and topological dynamics. In particular, the variational principles for polynomial topological entropy in zero entropy systems, mean dimensions in infinite entropy systems, and preimage entropy-like quantities in non-invertible dynamical systems are obtained.