Variational Principle for Relative Tail Pressure

Abstract

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.

Relative Tail Pressure with Subadditive Potentials

“…The first Lemma shows the upper semi-continuity of the conditional entropy, which can be found in many references, for instance [5,7]. The second one is Lemma 5 in [26], which shows the intrinsic connection relating the relative entropy with the relative tail pressure even in the general additive case.…”

“…In this section, we recall some basic notions of the relative measure-theoretic entropies for bundle random transformations [24,26]. For a general theory of random dynamical systems, we refer to [24,32,33].…”

“…For each ω, a standard argument shows that the sequence log P Φ Θ (ω, n, R, Q) is subadditive. By replacing the function S n f in Lemma 3.1 of Reference [26] with f n , one can easily get the following Lemma, which provides the basic measurable property needed. In fact, for any measurable function g on E , this result also holds.…”

“…be the open random cover of E generated by V (see [26] for details). Denote by δ(ω) the Lebesgue number of the open cover S(ω) for each ω ∈ Ω.…”

“…Ma et al set up a relative tail maximal entropy principle [25] and a tail variational principle [26] for continuous random transformations by introducing the relative tail entropy and pressure via open random covers, both of the two quantities are proved to be conserved under principal extensions. The notions defined there, via random covers, can enable ones to investigate different fibers under a natural but more complex cover way.…”