Local entropy theory of a random dynamical system

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Journal Article
Memoirs of the American Mathematical Society, 2015, 223 (1199), pp. 1 - 106
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© 2014 by the American Mathematical Society. All rights reserved. In this paper we extend the notion of a continuous bundle random dynamical system to the setting where the action of ℝ or ℕ is replaced by the action of an infinite countable discrete amenable group. Given such a system, and a monotone sub-additive invariant family of random continuous functions, we introduce the concept of local fiber topological pressure and establish an associated variational principle, relating it to measure-theoretic entropy. We also discuss some variants of this variational principle. We introduce both topological and measure-theoretic entropy tuples for continuous bundle random dynamical systems, and apply our variational principles to obtain a relationship between these of entropy tuples. Finally, we give applications of these results to general topological dynamical systems, recovering and extending many recent results in local entropy theory.
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