Admissibility and Hyperbolicity

1. Front Matter

   Pages i-ix

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2. Introduction

   • Luís Barreira, Davor Dragičević, Claudia Valls

   Pages 1-10

3. Exponential Contractions

   • Luís Barreira, Davor Dragičević, Claudia Valls

   Pages 11-32

4. Exponential Dichotomies: Discrete Time

   • Luís Barreira, Davor Dragičević, Claudia Valls

   Pages 33-54

5. Exponential Dichotomies: Continuous Time

   • Luís Barreira, Davor Dragičević, Claudia Valls

   Pages 55-74

6. Admissibility: Further Developments

   • Luís Barreira, Davor Dragičević, Claudia Valls

   Pages 75-106

7. Applications of Admissibility

   • Luís Barreira, Davor Dragičević, Claudia Valls

   Pages 107-137

8. Back Matter

   Pages 139-145

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This book gives a comprehensive overview of the relationship between admissibility and hyperbolicity. Essential theories and selected developments are discussed with highlights to applications. The dedicated readership includes researchers and graduate students specializing in differential equations and dynamical systems (with emphasis on hyperbolicity) who wish to have a broad view of the topic and working knowledge of its techniques. The book may also be used as a basis for appropriate graduate courses on hyperbolicity; the pointers and references given to further research will be particularly useful.

The material is divided into three parts: the core of the theory, recent developments, and applications. The first part pragmatically covers the relation between admissibility and hyperbolicity, starting with the simpler case of exponential contractions. It also considers exponential dichotomies, both for discrete and continuous time, and establishes corresponding results buildingon the arguments for exponential contractions. The second part considers various extensions of the former results, including a general approach to the construction of admissible spaces and the study of nonuniform exponential behavior. Applications of the theory to the robustness of an exponential dichotomy, the characterization of hyperbolic sets in terms of admissibility, the relation between shadowing and structural stability, and the characterization of hyperbolicity in terms of Lyapunov sequences are given in the final part. 

• Exponential Dichotomies
• Perron
• hyperbolicity
• admissibility
• dynamical systems
• differential equations
• Sequences of linear operators
• linear operators
• Nonuniqueness of solutions
• Robustness of hyperbolicity
• Lyapunov sequences
• Nonuniform Hyperbolicity
• Nonuniqueness of Solutions
• Admissible Spaces
• Robustness of Hyperbolicity
• Hyperbolic Sets
• Shadowing Property
• ordinary differential equations

“This book gives a comprehensive overview of the relationship between the notions of admissibility and hyperbolicity. … The book may also be used as a basis for appropriate graduate courses on hyperbolicity; the pointers and references given to further research will be particularly useful.” (Kazuhiro Sakai, zbMATH 1405.37002, 2019)

• Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

  Luís Barreira, Claudia Valls

• Department of Mathematics, University of Rijeka, Rijeka, Croatia

  Davor Dragičević

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