Abstract
We introduce the relative tail pressure to establish a variational principle for continuous bundle random dynamical systems. We also show that the relative tail pressure is conserved by the principal extension.
Highlights
The notion of topological pressure for the potential was introduced by Ruelle [1] for expansive dynamical systems
Walters [2] generalized it to the general case and established the classical variational principle, which states that the topological pressure is the supremum of the measure-theoretic entropy together with the integral of the potential over all invariant measures
Ledrappier [6] presented a variational principle between the topological tail entropy and the defect of upper semi-continuity of the measure-theoretic entropy on the cartesian square of the dynamical system involved, and proved that the tail entropy is an invariant under any principal extension
Summary
The notion of topological pressure for the potential was introduced by Ruelle [1] for expansive dynamical systems. Ledrappier [6] presented a variational principle between the topological tail entropy and the defect of upper semi-continuity of the measure-theoretic entropy on the cartesian square of the dynamical system involved, and proved that the tail entropy is an invariant under any principal extension. Kifer and Weiss [7] introduced the relative tail entropy for continuous bundle random dynamical systems (RDSs) by using the open covers and spanning subsets and deduced the equivalence between the two notions. We obtain a variational inequality, which shows that the defect of the upper semi-continuity of the relative measure-theoretic entropy of any invariant measure together with the integral of the random continuous potential in the product RDS cannot exceed the relative tail pressure of the original RDS.
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