Nonuniform Stationary Measure of the Invariant Unstable Foliation in Hamiltonian and Fluid Mixing Systems

This article demonstrates the existence of a nonuniform stationary measure (referred to as the w-invariant measure) associated with the statistical space-filling properties of the unstable invariant foliation in 2D differentiable area-preserving systems and derives a sequence of analytical approximations for it. In fluid-dynamic problems, the w-measure coincides with the intermaterial contact area density.