Abstract
A central role in the variational principle of the measure preserving transformations is played by the topological pressure. We introduce subadditive pre-image topological pressure and pre-image measure-theoretic entropy properly for the random bundle transformations on a class of measurable subsets. On the basis of these notions, we are able to complete the subadditive pre-image variational principle under relatively weak conditions for the bundle random dynamical systems.
Highlights
The topological pressure and variational principle play an important role in statistical mechanics, ergodic theory, and dynamical systems [1,2,3,4,5,6,7,8]
We present the notion of subadditive pre-image topological pressure Ppre ( T, Φ) for continuous random dynamical systems
We set up a variational principle for subadditive pre-image topological pressure for continuous bundle random dynamical systems
Summary
The topological pressure and variational principle play an important role in statistical mechanics, ergodic theory, and dynamical systems [1,2,3,4,5,6,7,8]. Ppre ( T, f ) = sup{ hpre, μ ( T ) + μ( f ) : μ ∈ M( T )}, where hpre, μ ( T ) is the pre-image measure-theoretic entropy, M( T ) is the set of T-invariant measures on the compact metric space X. We present the notion of subadditive pre-image topological pressure Ppre ( T, Φ) for continuous random dynamical systems. As for the case of −∞, the subadditive pre-image topological pressure is tightly relative to the limit function Φ∗ (μ), which is −∞ for all invariant measure μ. The pre-image measure-theoretic (relativized) entropy hpre, μ ( T ) of the RDS T with respect to μ ∈ M1P (E , T ) is defined by the formula hpre, μ ( T ) = sup hpre, μ ( T, Q),. We will use these results directly and not give these proofs since it seems that there is no different from the arguments in [30] except changing the symbols and restricting to the measurable subset E
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