matrices are elliptic.

Define $$F_I^\infty  = \{ f \in {\rm Diff}^\infty_+({\mathbb{T}}^1); \; \rho(f) \in {\mathbb{R}}{\smallsetminus}{\mathbb{Q}}\}$$ where $\rho(f)$ denotes the rotation number of $f$. We consider maps $A \in C^\infty_{\neq 0}({\mathbb{T}}^1,{\mathbb{G}})$, that is, smooth maps that are not homotopic to a constant matrix. The set $ \overline{F_I^\infty} \times  C^\infty_{\neq 0}({\mathbb{T}}^1,{\mathbb{G}})$ is a Baire space with the $C^\infty$-topology. Then:

There is a dense $G_\delta$ set $G \subset \overline{F_I^\infty} \times  C^\infty_{\neq 0}({\mathbb{T}}^1,{\mathbb{G}})$ of cocycles $(f,A)$ such that $f$ is uniquely ergodic and ${\mathit{LE}}(A, f,\mu_f)=0$ with $\mu_f$ the unique invariant probability of $f$.

Here the absence of uniform hyperbolicity is granted by the fact that the cocycle is not homotopic to identity – see [@HermanMethode proposition 4.2].

The set of smooth maps that are homotopic to a constant matrix is denoted by $C^\infty_{0}({\mathbb{T}}^1,{\mathbb{G}})$. Then:

There exists a set $F \subset F_I^\infty \times C^\infty_{0}({\mathbb{T}}^1,{\mathbb{G}})$ whose $C^\infty$ closure $\overline{F}$ is $C^0$ dense in the subset of non-uniformly hyperbolic cocycles in $F_I^\infty \times C^\infty_{0}({\mathbb{T}}^1,{\mathbb{G}})$, and such that there is a $C^\infty$ dense $G_\delta$ set $G \subset \overline{F}$ of cocycles $(f,A)$ such that $f$ is uniquely ergodic and ${\mathit{LE}}(A, f,\mu_f)=0$ with $\mu_f$ the unique invariant measure of $f$.

The discrete case
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We return to the measurable case 