This monograph presents new insights into the perturbation theory of dynamical systems based on the Gromov-Hausdorff distance. In the first part, the authors introduce the notion of Gromov-Hausdorff distance between compact metric spaces, along with the corresponding distance for continuous maps, flows, and group actions on these spaces. They also focus on the stability of certain dynamical objects like shifts, global attractors, and inertial manifolds. Applications to dissipative PDEs, such as the reaction-diffusion and Chafee-Infante equations, are explored in the second part. This text will be of interest to graduates students and researchers working in the areas of topological dynamics and PDEs.
Author(s): Jihoon Lee, Carlos Morales
Series: Frontiers in Mathematics
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 168
City: Cham
Preface
Contents
Part I Abstract Theory
1 Gromov-Hausdorff Distances
1.1 Introduction
1.2 Preliminary Facts
1.3 Gromov-Hausdorff Distance for Metric Spaces
1.4 Precompactness, Completeness, and a Variational Principle
1.5 A Variational Principle for the Gromov-Hausdorff Distance
1.6 Distances Characterizing Homeomorphic Spaces
1.7 C0-Gromov-Hausdorff Distance for Dynamical Systems
1.8 Gromov-Hausdorff Space of Continuous Maps
Exercises
2 Stability
2.1 Introduction
2.2 Definitions and Statement of Main Results
2.3 Proof of Theorem 2.1
2.4 Isometric Stability: Proof of Theorem 2.4
2.5 Proof of Theorem 2.5
2.6 Gromov-Hausdorff Stability for Group Actions
2.7 Gromov-Hausdorff Stability of Global Attractors
Exercises
3 Continuity of the Shift Operator
3.1 Introduction
3.2 Preliminary Facts
3.3 Proof of Theorem 3.1
3.4 Application to Stability Theory
Exercises
4 Shadowing from the Gromov-Hausdorff Viewpoint
4.1 Introduction
4.2 Definitions, Statement of Main Results, and Proofs
Exercises
Part II Applications to PDEs
Introduction
5 GH-Stability of Reaction-Diffusion Equations
5.1 Introduction
5.2 Proof of Theorem 5.1
5.3 Proof of Theorem 5.2
6 Stability of Inertial Manifolds
6.1 Introduction
6.2 Proof of Theorem 6.2
6.3 Proof of Theorem 6.3
7 Stability of Chafee-Infante Equations
7.1 Introduction
7.2 L-Morse-Smale and Equivalence of Global Attractors
7.3 Continuity of Global Attractors
7.4 Geometric Stability
References
