Nonremovable Zero Lyapunov Exponents

Skew products over a Bernoulli shift with a circle fiber are studied. We prove that in the space of such products there exists a nonempty open set of mappings each of which possesses an invariant ergodic measure with one of the Lyapunov exponents equal to zero. The conjecture that the space of $C^2$-diffeomorphisms of the $3$-dimensional torus into itself has a similar property is discussed.