Stability of Existence of Nonhyperbolic Measures for $C^1$-Diffeomorphisms

In the space of diffeomorphisms of an arbitrary closed manifold of dimension $\ge3$, we construct an open set such that each diffeomorphism in this set has an invariant ergodic measure with respect to which one of the Lyapunov exponents is zero. These diffeomorphisms are constructed to have a partially hyperbolic invariant set on which the dynamics is conjugate to a soft skew product with fiber the circle. It is the central Lyapunov exponent that proves to be zero in this case, and the construction is based on an analysis of properties of the corresponding skew products.