  In this article we study random tower maps driven by an ergodic automorphism.
We prove quenched exponential correlations decay for tower maps admitting
exponential tails. Our technique is based on constructing suitable cones of
functions, defined on the random towers, which contract with respect to the
Hilbert metric under the action of appropriate transfer operators. We apply our
results to obtain quenched exponential correlations decay for several non-iid
random dynamical systems including small random perturbations of Lorenz maps
and Axiom A attractors.
