Algebraic function dynamical system

Author: Zheng Jianhua
Publisher:
Publish Date: 2006-06-01
Features: This book primarily introduces dynamics beyond the iteration of meromorphic functions, and to some extent, also covers the dynamics of rational functions. The content of this book not only includes the fundamental theories in complex dynamics but also introduces a large number of new results. It strives to present these new results from different perspectives or viewpoints, either simplifying the originally complex proofs or providing alternative proofs. Additionally, relevant prerequisite knowledge is interspersed throughout the chapters to facilitate reading. The main content of this book includes the following aspects: the existence of periodic points of meromorphic functions, providing quantitative estimates for the number of proper periodic points; the conjugacy problem of self-maps on hyperbolic regions and M?bius transformations; the properties and existence of various periodic domains and wandering domains; the characteristics of Julia sets, such as the distribution of Julia sets, the existence of single-point branches and immersed branches, uniform completeness, and Lebesgue measure; the stability and structural stability of meromorphic function families; the Hausdorff dimension of Julia sets, particularly for representative classes of meromorphic function Julia sets; finally, a class of measurable dynamics of meromorphic functions, determining the existence of ergodic Gibbs invariant measures and their Hausdorff dimensions, as well as the relationships between Gibbs measure entropy and Lyapunov exponents.