  We consider a family of skew-products of the form $(Tx, g_x(t)) : X \times
\mathbb{R} \to X \times \mathbb{R}$ where $T$ is a continuous expanding Markov
map and $g_x : \mathbb{R} \to \mathbb{R}$ is a family of homeomorphisms of
$\mathbb{R}$. A function $u: X \to \mathbb{R}$ is said to be an invariant graph
if $\mathrm{graph}(u) = \{(x,u(x)) \mid x\in X\}$ is an invariant set for the
skew-product; equivalently if $u(T(x)) = g_x(u(x))$. A well-studied problem is
to consider the existence, regularity and dimension-theoretic properties of
such functions, usually under strong contraction or expansion conditions (in
terms of Lyapunov exponents or partial hyperbolicity) in the fibre direction.
Here we consider such problems in a setting where the Lyapunov exponent in the
fibre direction is zero on a set of periodic orbits. We prove that $u$ either
has the structure of a `quasi-graph' (or `bony graph') or is as smooth as the
dynamics, and we give a criteria for this to happen.
