Non-uniform stationary measure properties of chaotic area-preserving dynamical systems

This article shows the existence of a non-uniform stationary measure (referred to as the w-invariant measure) associated with the space-filling properties of the unstable manifold and characterizing some statistical properties of chaotic two-dimensional area-preserving systems. The w-invariant measure, which differs from the ergodic measure and is non-uniform in general, plays a central role in the statistical characterization of chaotic fluid mixing systems, since several properties of partially mixed structures can be expressed as ensemble averages over the w-invariant measure. A closed-form expression for the w-invariant density is obtained for a class of mixing systems topologically conjugate with the linear toral automorphism. The physical implications in the theory of fluid mixing, and in the statistical characterization of chaotic Hamiltonian systems, are discussed.