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 $\mathrm{graph}(u) = \{(x,u(x)) \mid x\in X\}$ 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.Comment: 21 pages, 2 figure

The HeliPaD:A parsed corpus of Old Saxon

This short paper introduces the HeliPaD, a new parsed corpus of Old Saxon (Old Low German). It is annotated according to the standards of the Penn Corpora of Historical English, enriched with lemmatization and additional morphological attributes as well as textual and metrical annotation. This paper provides an overview of its main features and compares it to existing resources such as the Deutsch Diachron Digital version of the Old Saxon Heliand as part of the Referenzkorpus Altdeutsch. It closes with a roadmap for planned future expansions.publishe