Dynamics Beyond Uniform Hyperbolicity

1. Front Matter

   Pages i-xviii

   PDF

2. Hyperbolicity and Beyond

   Pages 1-11

3. One-Dimensional Dynamics

   Pages 13-24

4. Homoclinic Tangencies

   Pages 25-54

5. Hénon-like Dynamics

   Pages 55-95

6. Non-Critical Dynamics and Hyperbolicity

   Pages 97-106

7. Heterodimensional Cycles and Blenders

   Pages 107-121

8. Robust Transitivity

   Pages 123-146

9. Stable Ergodicity

   Pages 147-155

10. Robust Singular Dynamics

    Pages 157-188

11. Generic Diffeomorphisms

    Pages 189-212

12. SRB Measures and Gibbs States

    Pages 213-251

13. Lyapunov Exponents

    Pages 253-275

14. Back Matter

    Pages 277-384

    PDF

What is Dynamics about? In broad terms, the goal of Dynamics is to describe the long term evolution of systems for which an "infinitesimal" evolution rule is known. Examples and applications arise from all branches of science and technology, like physics, chemistry, economics, ecology, communications, biology, computer science, or meteorology, to mention just a few. These systems have in common the fact that each possible state may be described by a finite (or infinite) number of observable quantities, like position, velocity, temperature, concentration, population density, and the like. Thus, m the space of states (phase space) is a subset M of an Euclidean space M . Usually, there are some constraints between these quantities: for instance, for ideal gases pressure times volume must be proportional to temperature. Then the space M is often a manifold, an n-dimensional surface for some n < m. For continuous time systems, the evolution rule may be a differential eq- tion: to each state x G M one associates the speed and direction in which the system is going to evolve from that state. This corresponds to a vector field X(x) in the phase space. Assuming the vector field is sufficiently regular, for instance continuously differentiable, there exists a unique curve tangent to X at every point and passing through x: we call it the orbit of x.

• attractor
• diffeomorphism
• dynamical systems
• hyperbolicity
• physical measure
• robustness

From the reviews:

"... Nonuniformly hyperbolic phenomena are a central theme in current research in dynamical systems theory. The attractive book under review is meant as a guide for students as well as established researchers to explore these new and exciting ideas.  ...

The book is well written by recognized expers who have made significant contributions to their subject. They have wisely chosen to make the various sections of their book essentially self-contained. Thus, the narrative is suitable for cover-to-cover exploration or more concentrated study of specific topics. While definiotions and theorems are stated precisely, only outlines of most of the proofs are given; the reader is referred (via a bibliography with 466 entries) to the research literature for details. This practice sets the style for the book: The main ideas are front and center, where they should be for the current state of the subject to come rapidly into focus. Students - young and old - will find here a broad and insightful overview of recent results in modern dynamical systems theory, where many open problems and future directions for research are discussed. ...

The authors have good reason to believe that many readers will be inspired by their excellent book."

Carmen Chicone, Univ. of Missouri-Columbia, Siam Review, Issue 47, No. 4, 2005

"The notion of uniform hyperbolicity led to the remarkable advances in the theory of dynamical systems in the 60ies and 70ies of the 20th century … . The book is a welcome reference source for researchers and graduate students who work in or just want to get impression on this important and rapidly developing area of dynamical systems." (Yuri Kifer, Zentralblatt MATH, Vol. 1060, 2005)

• Institut de Mathématiques de Bourgogne, UMR 5584 du CNRS, Université de Bourgogne, Dijon Cedex, France

  Christian Bonatti

• Departamento de Matemática, PUC-Rio, Gávea — Rio de Janeiro, Brazil

  Lorenzo J. Díaz

• IMPA, Jardim Botânico — Rio de Janeiro, Brazil

  Marcelo Viana

Policies and ethics