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

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.

21 pages, 2 figures

References in corpus (2)

• On the dimension of the graph of the classical Weierstrass function
• Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors