Skew Product Attractors and concavity

TL;DR

This paper introduces a new approach to skew product attractors that minimizes base space assumptions and explores the effects of fiber map concavity on contraction properties, even with noninvertible base maps.

Contribution

It presents a novel framework for analyzing skew product attractors without assuming invariance and demonstrates fiber contraction independence from base map invertibility when fiber maps are concave.

Findings

Avoids unnecessary base space structures in attractor analysis

Shows fiber contraction is independent of base map invertibility with concave fibers

Addresses the phenomenon of vanishing attractors in noninvertible base maps

Abstract

We propose an approach to the attractors of skew products that tries to avoid unnecessary structures on the base space and rejects the assumption on the invariance of an attractor. When nonivertible maps in the base are allowed, one can encounter the mystery of the vanishing attractor. In the second part of the paper, we show that if the fiber maps are concave interval maps then contraction in the fibers does not depend on the map in the base.