Abstract:
We investigate the dynamics of certain homeomorphisms
$F\colon\mathbb{T}^2\to\mathbb{T}^2$ of the form $ F(x,y)=(x+\omega,h(x)+f(y)), $
where $\omega\in\mathbb{R}\setminus \mathbb{Q}$, $f\colon \mathbb{T}\to\mathbb{T}$ is a
circle diffeomorphism with a unique (and thus neutral) fixed point
and $h\colon \mathbb{T}\to\mathbb{T}$ is a function which is zero outside a small
interval. We show that such a map can display a non-uniformly
hyperbolic behavior: (small) negative fibred Lyapunov exponents for
a.e. $(x,y)$ and an attracting non-continuous invariant graph. We
apply this result to (projective) $\mathrm{SL}(2,\mathbb{R})$-cocycles $G\colon
(x,u)\mapsto (x+\omega,A(x)u)$ with $A(x)=R_{\phi(x)}B$, where
$R_\theta$ is a rotation matrix and $B$ is a parabolic matrix, to
get examples of non-uniformly hyperbolic cocycles (homotopic to the
identity) with perturbatively small Lyapunov exponents.
Key words and phrases:Lyapunov exponents, quasi-periodic forcing, nonuniform hyperbolicity, cocycles.
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