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Über dieses Buch

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 building on 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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

As already noted in the preface, the main objective of this book is to give a fairly broad overview of the relation between admissibility and hyperbolicity. In this chapter we describe in a pragmatic manner the origins of the theory and we give a brief overview of the contents of the book.
Luís Barreira, Davor Dragičević, Claudia Valls

Chapter 2. Exponential Contractions

In this chapter we present the main results of the admissibility theory in the simpler case of exponential contractions, for both discrete and continuous time. This allows us to give a first introduction to the relation between hyperbolicity and admissibility without the technical complications caused by the existence of contraction and expansion in an exponential dichotomy. The results presented here are generalized in Chapters 3 and 4 to exponential dichotomies, respectively, for discrete and continuous time.
Luís Barreira, Davor Dragičević, Claudia Valls

Chapter 3. Exponential Dichotomies: Discrete Time

In this chapter we start discussing the admissibility theory in the general case of exponential dichotomies. The objective is the same—to characterize the notion of an exponential dichotomy in terms of an admissibility property. The arguments build substantially on those in Chapter 2, although there are various technical difficulties that need to be overcome to treat the general case. The major difficulty consists of showing that an admissibility property implies the existence of contracting and expanding directions, with invertibility along the unstable direction. In this chapter we consider only the case of discrete time. In Chapter 4 we develop a corresponding theory for continuous time.
Luís Barreira, Davor Dragičević, Claudia Valls

Chapter 4. Exponential Dichotomies: Continuous Time

This chapter is dedicated to the study of the admissibility theory for exponential dichotomies in continuous time. Again, the arguments build on those in Chapter 2, up to substantial technical complications. To the possible extent, we follow the path of Chapter 3. In particular, we consider both a two-sided and a one-sided dynamics given by an evolution family.
Luís Barreira, Davor Dragičević, Claudia Valls

Chapter 5. Admissibility: Further Developments

In this chapter we consider various extensions of the results in the former chapters. In particular, we develop a general approach to the problem of constructing pairs of Banach spaces whose admissibility property can be used to characterize an exponential dichotomy. This generalizes and unifies some of the results in the former chapters. Moreover, we discuss what we call Pliss type theorems. These results deal with a weaker form of admissibility on the line not requiring the uniqueness condition and guarantee the existence of exponential dichotomies on both the positive and negative half-lines. Finally, we introduce the more general notion of a nonuniform exponential dichotomy and again we characterize it in terms of an appropriate admissibility property also for maps and flows.
Luís Barreira, Davor Dragičević, Claudia Valls

Chapter 6. Applications of Admissibility

In this chapter we describe various applications of the results in the former chapters. In particular, we establish the robustness property of an exponential dichotomy by showing that its stability persists under sufficiently small linear perturbations. Moreover, we develop a characterization of hyperbolic sets in terms of an appropriate admissibility property for both maps and flows. Furthermore, we discuss applications of the Pliss type theorems to shadowing and its relation to structural stability. Finally, we obtain a complete characterization of an exponential dichotomy in terms of the existence of a Lyapunov sequence. We do not strive to present the most general results so that one can avoid accessory technicalities.
Luís Barreira, Davor Dragičević, Claudia Valls

Backmatter

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