1 \\ 1 & 0 \end{pmatrix},
    \quad \theta \in {\mathbb{T}}^d, \ \lambda \in {\mathbb{R}}$$ If $V(\theta)$ is a non-constant trigonometrical polynomial, and the dynamics in the base is restricted to real analytic maps in a neighborhood of the unit polydisc in ${\mathbb{C}}^d$, then Herman [@HermanMethode] showed that there exists a positive lower bound on the exponent, provided $\lambda$ is greater than some $\lambda_0$.

In our setting, we can ask:

Assume $A \colon X \to {\mathbb{G}}$ is a $C^1$ map that assumes both elliptic and hyperbolic values. (A concrete interesting example in the torus $X={\mathbb{T}}^d$ is $A=S_{V,\lambda}$ given by with $V(\theta) = \cos \theta_1$ and $\lambda \gg 1$.) When is it possible to find a volume-preserving $C^1$ map $T \colon X \to X$ such that ${\mathit{LE}}(A,T)$ is exactly zero?

We mention here two results obtained by Herman proving abundance of zero exponents in the absence of uniform hyperbolicity for smooth cocycles above uniquely ergodic diffeomorphisms of the circle. (Here the exponents are computed above the unique invariant probability measure.) The results are based on Baire category arguments and the method used is to approximate the base dynamics by periodic maps and concentrate the measure on orbits above which the product of 