Dynamics Beyond Uniform Hyperbolicity: A Global Geometric and Probabilistic Perspective

ISBN-13: 9783642060410 / Angielski / Miękka / 2010 / 384 str.

Christian Bonatti;Lorenzo J. Díaz;Marcelo Viana

Opis

The notion of uniform hyperbolicity, introduced by Steve Smale in the early sixties, unified important developments and led to a remarkably successful theory for a large class of systems: uniformly hyperbolic systems often exhibit complicated evolution which, nevertheless, is now rather well understood, both geometrically and statistically.

Another revolution has been taking place in the last couple of decades, as one tries to build a global theory for "most" dynamical systems, recovering as much as possible of the conclusions of the uniformly hyperbolic case, in great generality.

This book aims to put such recent developments in a unified perspective, and to point out open problems and likely directions for further progress. It is aimed at researchers, both young and senior, willing to get a quick, yet broad, view of this part of dynamics. Main ideas, methods, and results are discussed, at variable degrees of depth, with references to the original works for details and complementary information.

Recenzje

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)

Spis treści

Hyperbolicity and Beyond.- One-Dimensional Dynamics.- Homoclinic Tangencies.- Hénon-like Dynamics.- Non-Critical Dynamics and Hyperbolicity.- Heterodimensional Cycles and Blenders.- Robust Transitivity.- Stable Ergodicity.- Robust Singular Dynamics.- Generic Diffeomorphisms.- SRB Measures and Gibbs States.- Lyapunov Exponents.

Blurb

In broad terms, the goal of dynamics is to describe the long-term evolution of systems for which an "infinitesimal" evolution rule, such as a differential equation or the iteration of a map, is known.

The notion of uniform hyperbolicity, introduced by Steve Smale in the early sixties, unified important developments and led to a remarkably successful theory for a large class of systems: uniformly hyperbolic systems often exhibit complicated evolution which, nevertheless, is now rather well understood, both geometrically and statistically.

Another revolution has been taking place in the last couple of decades, as one tries to build a global theory for "most" dynamical systems, recovering as much as possible of the conclusions of the uniformly hyperbolic case, in great generality.

This book aims to put such recent developments in a unified perspective, and to point out open problems and likely directions for further progress. It is aimed at researchers, both young and senior, willing to get a quick, yet broad, view of this part of dynamics. Main ideas, methods, and results are discussed, at variable degrees of depth, with references to the original works for details and complementary information.

The 12 chapters are organised so as to convey a global perspective of this field, but they have been kept rather independent, to allow direct access to specific topics. The five appendices cover important complementary material.