Invariant geometric properties of class of 3D chaotic flows
The paper deals with \(C^{\infty}\) volume-preserving diffeomorphisms \(\Phi = \mathcal H^{-1} \circ \mathcal A \circ \mathcal H\) of the \(3\)-dimensional torus \(T^3\) topologically conjugate to a linear map \(\mathcal A(x)\) where \(\circ\) indicates composition and \(\mathcal H(x)\) is a generic \(C^{\infty}\) volume-preserving diffeomorphism. The authors determine the global geometrical properties and the resulting pointwise characterization of the evolution of curves and surfaces advected by the diffeomorphism \(\Phi.\) The measure-theoretical properties associated with the asymptotic evolution of so-called material lines and surfaces are analyzed. The connection between the theory of the asymptotic properties of the diffeomorphism \(\Phi\) and the properties of physically realizable \(3\)-dimensional chaotic flows is discussed.
scientific article; zbMATH DE number 1496380
scientific article