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It is well-known that a $C^2$ Axiom A basic set $\Lambda$ is an attractor if and only if the topological pressure with respect to the negative unstable log-determinant vanishes (see Theorem 4.11 in Bowen's book "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms"). This, in turn, is equivalent to the existence of an SRB measure on $\Lambda$. My question is whether an analogous result exists for uniformly hyperbolic sets of random dynamical systems, in particular for the case when the uniformly hyperbolic set arises by a small random perturbation of an Axiom A basic set of a diffeomorphism (as studied in the paper "Random Perturbations of Axiom A Basic Sets" by P.-D. Liu). In this particular situation, it would also be interesting to know whether the existence of an SRB measure for the random dynamical system implies that the Axiom A basic set we started with is an attractor (the converse is proved in Liu's paper). Note that by "attractor" I mean that an open set of initial states is attracted to the set.

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