Connected Sets in Global Bifurcation Theory

by Boris Buffoni, John Toland

Overview

This book explores the topological properties of connected and path-connected solution sets for nonlinear equations in Banach spaces, focusing on the distinction between these concepts. Building on Rabinowitz's dichotomy and classical results on Peano continua, the authors introduce "congestion points"—where connected sets fail to be weakly locally connected—and examine the extent to which their presence is compatible with path-connectedness. Through rigorous analysis and examples, the book provides new insights into global bifurcations.

Structured into seven chapters, the book begins with an introduction to global bifurcation theory and foundational concepts in set theory and metric spaces. Subsequent chapters delve into connectedness, local connectedness, and congestion points, culminating in the construction of intricate examples that highlight the complexities of solution sets. The authors' careful selection of material and fluent writing style make this work a valuable resource for PhD students and experts in functional analysis and bifurcation theory.

Product Details

ISBN-13:	9783031870507
Publisher:	Springer Nature Switzerland
Publication date:	04/30/2025
Series:	SpringerBriefs in Mathematics
Pages:	101
Product dimensions:	6.10(w) x 9.25(h) x (d)

About the Author

Boris Buffoni works at the Institute of Mathematics at EPFL (Ecole Polytechnique Fédérale de Lausanne, Switzerland), where he has taught since 1998. His doctorate, under the supervision of Charles Stuart at EPFL, focused on nonlinear problems in the presence of essential spectrum. He was a postdoctoral researcher at the University of Bath and the Scuola Normale Superiore in Pisa, and, from 1995 to 1998, a lecturer at the University of Bath. His research interests include calculus of variations, bifurcation theory, partial differential equations and applications to hydrodynamics. He is currently a senior scientist at EPFL.

John Toland is Emeritus Professor of Mathematics at the University of Bath where he was professor for 32 years before being appointed Director of the Isaac Newton Institute in Cambridge.

His PhD, under the supervision of Charles Stuart at Sussex University, was on global bifurcation theory for k-set-contractions after which, with collaborators, he developed topological methods to prove the existence of large amplitude solitary waves, including the famous singular Stokes-wave-of-greatest-height. Since then he has developed aspects of convex analysis, harmonic analysis, duality theory, Nash-Moser theory and variational methods, to address nonlinear problems arising in applications.

Table of Contents

- 1. Introduction.- 2. Set Theory Foundations.- 3. Metric Spaces.- 4. Types of Connectedness.- 5. Congestion Points.- 6. Decomposable and Indecomposable Continua.- 7. Pathological Examples.