Abstract
We consider a family of skew-products of the form where T is a continuous, expanding, locally eventually onto Markov map and is a family of homeomorphisms of . A function is said to be an invariant graph if is an invariant set for the skew-product; equivalently, 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 but expands except on a neighbourhood of these 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.
Original language | English |
---|---|
Article number | 2726 |
Journal | Nonlinearity |
Volume | 31 |
Issue number | 6 |
Early online date | 30 Apr 2018 |
DOIs | |
Publication status | Published - Jun 2018 |