Invariant geometric properties of a class of 3D chaotic flows

This article extends the analysis developed by Giona and Adrover [M. Giona, A. Adrover, Phys. Rev. Lett. 81 (1998) 3864] for 2D area-preserving diffeomorphisms to 3D volume-preserving C∞-diffeomorphisms of the 3D torus topologically conjugate to a linear map. The article analyzes the invariant geometric properties of vector dynamics and surface element evolution in 3D systems and provides an analytic expression for the probability measure describing pointwise statistical properties of the unstable foliations in the hyperbolic case. The convergence properties of this measure are addressed starting from the dynamics of surface elements. The application of the methods developed to physically realizable 3D chaotic flows such as ABC flow is discussed in detail.