We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, gives explicit examples of ergodic R-extensions of minimal locally Hamiltonian flows with non-degenerate saddles in genus two. More generally, given any symmetric irreducible permutation, we show that for almost every choice of lengths vector, the skew-product built over the IET with the given permutation and lengths vector given by a cocycle, with symmetric, logarithmic singularities, which is odd when restricted to each continuity subinterval is ergodic.

For a fixed frequency ω P R 2 zt0u we show the existence of Gevrey smooth Hamiltonians, arbitrarily close to a Kolmogorov non-degenerate analytic Hamiltonian, having a Lyapunov unstable elliptic equilibrium with frequency ω. In particular, the Hamiltonians thus constructed will be KAM-stable.

No abstract