Invariant multi-graphs in step skew-products

TL;DR

This paper investigates the structure of invariant sets in step skew-products with interval maps, revealing multi-graph formations and phase space decompositions into attracting and repelling regions with graph-like attractors.

Contribution

It introduces the concept of multi-graph structures in invariant sets and characterizes phase space decomposition in absorbing skew-products.

Findings

Invariant sets can be expressed as multi-graphs of functions.

Hyperbolic sets and ergodic hyperbolic measures support multi-graph structures.

Phase space decomposes into attracting and repelling double-strips with graph-based attractors and repellers.

Abstract

We study step skew-products over a finite-state shift (base) space whose fiber maps are $C^1$ injective maps on the unit interval. We show that certain invariant sets have a multi-graph structure and can be written graphs of one, two or more functions defined on the base. In particular, this applies to any hyperbolic set and to the support of any ergodic hyperbolic measure. Moreover, within the class of step skew-products whose interval maps are 'absorbing', open and densely the phase space decomposes into attracting and repelling double-strips such that their attractors and repellers are graphs of one single-valued or bi-valued continuous function almost everywhere, respectively.