Loosely Bernoulli zero exponent measures for elliptic matrix cocycles

Abstract: For an open and dense subset of elliptic ${\rm SL}(2,\mathbb R)$ matrix cocycles, we construct a family of loosely Bernoulli ergodic measures with zero top Lyapunov exponent. This provides a counterpart to a classical result by Furstenberg. The construction gives also an $\bar f$-connected set of measures with these properties whose entropies vary continuously from zero to almost the maximal possible value. We also obtain an analogous result for an open class of nonhyperbolic step skew products with $\mathbb S^1$ diffeomorphism fiber maps. Our approach combines substitution schemes between finite letter alphabets and differentiable dynamics.