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

Charles Walkden, Thomas Withers

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    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 languageEnglish
    Article number2726
    JournalNonlinearity
    Volume31
    Issue number6
    Early online date30 Apr 2018
    DOIs
    Publication statusPublished - Jun 2018

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