Hyperbolicity of smooth cocycles over flows with invariant ergodic measure

Completely Squashable Smooth Ergodic Cocycles over Irrational Rotations

Let α be an irrational number and the trasformation Tx 7→ x+ α mod 1, x ∈ [0, 1), represent an irrational rotation of the unit circle. We construct an ergodic and completely squashable smooth real extension, i.e. we find a real analytic or k time continuously differentiable real function F such that for every λ 6= 0 there exists a commutor Sλ of T such that F ◦Sλ is T -cohomologous to λφ and th...

Uniform Szego cocycles over strictly ergodic subshifts

We consider ergodic families of Verblunsky coefficients generated by minimal aperiodic subshifts. Simon conjectured that the associated probability measures on the unit circle have essential support of zero Lebesgue measure. We prove this for a large class of subshifts, namely those satisfying Boshernitzan’s condition. This is accomplished by relating the essential support to uniform convergenc...

On SL(2, R) valued smooth proximal cocycles and cocycles with positive Lyapunov exponents over irrational rotation flows

Consider the class of C-smooth SL(2,R) valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: ∣ α− pn qn ∣ ∣ ≤ Cexp(−q n ), where C > 0 a...

On the space of ergodic invariant measures of unipotent flows

Let G be a Lie group and Γ be a discrete subgroup. We show that if {μn} is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G.

Ergodic Theory of Markov Chains Admitting an Infinite Invariant Measure.