圆映射动力学 第二版

English | 2024 | ISBN-10: 3031674944 | 476 pages| Epub PDF (True) | 51 MB

This book explores recent developments in the dynamics of invertible circle maps, a rich and captivating topic in one-dimensional dynamics. It focuses on two main classes of invertible dynamical systems on the circle: global diffeomorphisms and smooth homeomorphisms with critical points. The latter is the book's core, reflecting the authors' recent research interests. Organized into four parts and 14 chapters, the content covers rigid rotations, circle homeomorphisms, and the concept of rotation number in the first part. The second part delves into circle diffeomorphisms, presenting classical results. The third part introduces multicritical circle maps―smooth homeomorphisms of the circle with a finite number of critical points. The fourth and final part centers on renormalization theory, analyzing the fine geometric structure of orbits of multicritical circle maps. Complete proofs for several fundamental results in circle dynamics are provided throughout. The book concludes with a list of open questions.

这本书探讨了逆向圆映射的一维动力学的最新发展，这是一个在单一维度动力学中既丰富又引人入胜的主题。它关注两种主要类别的圆上可逆动力系统：全局微分同胚和有临界点的光滑同胚。后者是本书的核心，反映了作者们近期的研究兴趣。全书分为四部分和14章的内容涵盖了刚性旋转、圆映射以及旋转数的概念在第一部分中。第二部分深入到圆映射领域，介绍了经典结果。第三部分介绍了多临界圆映射——有有限个临界点的光滑同胚的圆映射。最后一部分集中分析了多重临界圆映射轨道的精细几何结构。全书中提供了几个基本圈上动力学结果的完整证明。本书以一个开放问题清单结束。

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