A fluctuation theorem in a random environment

The space of the random events Ω will be supposed a space of sequences of finitely many symbols ω = {ωj}j=−∞, for instance a sequence of spins ωj = ±1, with the usual metric d(ω,ω) = ∑ j |ωj − ω j|2, and with τ the shift to the left: (τω)j = ωj+1. A “reflection” operation will be defined on Ω as ω←→ωT , with (ω )j = ω−j. The probability distribution P will be a τ–invariant and reflection–invariant mixing process on Ω, for instance a Bernoulli shift or a Markov process with symmetric transition probabilities or a suitably symmetric general Gibbs distribution on Ω. Reflection–invariant means that P (E) ≡ P (E ) for every Borel set E, where E is the image of E under the reflection ω←→ωT . The manifold M will be a torus Tm and fω will be supposed to be close to a map f which is a “linear hyperbolic torsion” of Tm, independent of the environment, i.e. close to a map which is defined by the action of a hyperbolic matrix with integer entries and determinant ±1 on the torus. For instance the point x ∈ M can be visualized as a pair of clock arms x = φ = (φ1, φ2)