On entropy and pressure via preimage structure for random dynamical systems

Abstract

In this paper, entropy, pressure and equilibrium states are investigated for noninvertible random dynamical systems via the preimage structure. For a continuous random dynamical system F on a compact metric space and an F-invariant measure μ, the folding entropy Fμ(F) and the preimage entropy hm,μ(F) are introduced, and it is shown that these two measure-theoretic versions of entropies coincide with each other if F has uniform separation of preimages. Some fundamental properties for hm,μ(F) such as Shannon-McMillan-Breiman Theorem, Kolmogorov-Sinai Theorem and upper semi-continuity are established. From the topological viewpoint, the preimage pressures Pm(F,⋅), Pp(F,⋅) and Pp˜(F,⋅) (resp. topological preimage entropy hm(F), hp(F) and hp˜(F)) are introduced via the preimage structure of single points. Some basic properties are discussed, and particularly, variational principles are obtained. For a C1 nondegenerate random partially hyperbolic dynamical system F on a closed Riemannian manifold, we propose the notions of metric stable entropy hm,μs(F) and stable pressures Pms(F,⋅) and Pps(F,⋅) (resp. topological stable entropies hms(F) and hps(F)) via the preimage structure of local stable manifolds. Several basic properties are obtained and variational principles are established particularly. At the end, as an application of variational principles, p-equilibrium states (resp. s-equilibrium states) and p-tangent functionals (resp. s-tangent functionals) are studied.