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The monograph is devoted to the study of functional equations with the transformed argument on the real line and on the unit circle. Such equations systematically arise in dynamical systems, differential equations, probabilities, singularities of smooth mappings, and other areas. The purpose of the book is to present modern methods and new results in the subject, with an emphasis on a connection between local and global solvability. The general concepts developed in the book are applicable to multidimensional functional equations. Some of the methods are presented for the first time in the monograph literature.
The book is addressed to graduates and researchers interested in dynamical systems, differential equations, operator theory, or the theory of functions and their applications.
List of contents
1 Implicit Functions.- 1.1 Formal solvability.- 1.2 Theorem on local solvability.- 1.3 Transformations of equations.- 1.4 Global solvability.- 1.5 Comments and references.- 2 Classification of One-dimensional Mappings.- 2.1 Wandering and non-wandering subsets.- 2.2 Mappings with wandering compact sets.- 2.3 Local structure of mappings at an isolated fixed point.- 2.4 Diffeomorphisms with isolated fixed points.- 2.5 One-dimensional flows and vector fields.- 2.6 Embedding problem and iterative roots.- 2.7 Comments and references.- 3 Generalized Abel Equation.- 3.1 Local solvability.- 3.2 Global solutions of equations with not more than one fixed point.- 3.3 Gluing method for linear equations with several fixed points.- 3.4 Comments and references.- 4 Equations with Several Transformations of Argument.- 4.1 Local solvability.- 4.2 Extension of solutions.- 4.3 Examples.- 4.4 Difference equations in Carleman classes.- 4.5 Comments and references.- 5 Linear Equations.- 5.1 Generalized linear Abel equation.- 5.2 Localization of obstacles to solvability.- 5.3 Equations with constant coefficients.- 5.4 Equation with affine transformations of argument.- 5.5 Comments and references.
About the author
Vadim Tkachenko was a Performance Engineer in at MySQL AB. As an expert in multithreaded programming and synchronization, his primary tasks were benchmarks, profiling, and finding bottlenecks. He also worked on a number of features for performance monitoring and tuning, and getting MySQL to scale well on multiple CPUs.
Summary
The monograph is devoted to the study of functional equations with the transformed argument on the real line and on the unit circle. Such equations systematically arise in dynamical systems, differential equations, probabilities, singularities of smooth mappings, and other areas. The purpose of the book is to present modern methods and new results in the subject, with an emphasis on a connection between local and global solvability. The general concepts developed in the book are applicable to multidimensional functional equations. Some of the methods are presented for the first time in the monograph literature.
The book is addressed to graduates and researchers interested in dynamical systems, differential equations, operator theory, or the theory of functions and their applications.
