Attracting graphs of skew products with non-contracting fiber maps

TL;DR

This paper investigates attracting graphs in step skew products with non-contracting fiber maps, establishing existence, topological properties, and convergence rates of attracting measures under weaker conditions.

Contribution

It introduces new topological conditions for attracting graphs in skew products, extending beyond traditional contraction assumptions.

Findings

Existence of attracting invariant graphs

Presence of globally attracting measures

Exponential convergence rates in specific cases

Abstract

We study attracting graphs of step skew products from the topological and ergodic points of view where the usual contracting-like assumptions of the fiber dynamics are replaced by weaker merely topological conditions. In this context, we prove the existence of an attracting invariant graph and study its topological properties. We prove the existence of globally attracting measures and we show that (in some specific cases) the rate of convergence to these measures is exponential.