Invariant graphs of a family of non-uniformly expanding skew products over Markov maps

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 $\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}(u)=\{(x,u(x))\mid x\in X\}$ 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.