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

## TL;DR

This paper studies invariant graphs in skew-product systems over Markov maps, showing they are either quasi-graphs or smooth, especially when the fiber Lyapunov exponent is zero on periodic orbits.

## Contribution

It introduces a new analysis of invariant graphs in non-uniformly expanding skew-products with zero fiber Lyapunov exponents, characterizing their structure and regularity.

## Key findings

- Invariant graphs are either quasi-graphs or smooth.
- Criteria established for the structure of invariant graphs.
- Analysis extends understanding of non-uniform hyperbolicity in skew-products.

## Abstract

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.