Singular-Hyperbolicity and Volume

Abstract

It is well known that a C ² dynamical system admitting a hyperbolic basic set with positive measure must be globally hyperbolic: see e.g. Bowen-Ruelle (Invent. Math. 29:181–202, 1975) and Bochi-Viana (Advances in Dynamical Systems, Cambridge University Press, Cambridge, 2004). The construction of the geometric Lorenz models, presented in Chap. 3, forces the divergence of the vector field to be strictly negative in a isolating neighborhood of the attractor. This feature is also present in the Lorenz system of equations (2.2) for the classical parameters. It is then trivial to show that the corresponding attractor has zero volume.

Using only the partial hyperbolic character of singular-hyperbolicity we show in Sect. 8.2, following (Alves et al. in Dyn. Syst. 22(3):249–267, 2007) for flows which are Hölder-C ¹, that either singular-hyperbolic attractors have zero volume or else the flow is globally hyperbolic, that is, an Anosov flow (without singularities).