One-dimensional Functional Equations [Hardcover]

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.

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.- 2.2.1 Strictly monotonic mappings without fixed points.- 2.2.2 The Abel and cohomological equations.- 2.2.3 Smooth and analytic solutions of a cohomological equation.- 2.3 Local structure of mappings at an isolated fixed point.- 2.3.1 Formal classification.- 2.3.2 Smooth classification.- 2.3.3 Analytic classification.- 2.4 Diffeomorphisms with isolated fixed points.- 2.4.1 Topological classification.- 2.4.2 Smooth classification of diffeomorphisms with a unique fixed point.- 2.4.3 Smooth classification of diffeomorphisms with several hyperbolic fixed points.- 2.4.4 Another approach to smooth classification.- 2.5 One-dimensional flows and vector fields.- 2.5.1 Classification of vector fields in a neighborhood of a singular point.- 2.5.2 Flows on the real line with hyperbolic fixed points.- 2.6 Embedding problem and iterative roots.- 2.6.1 Mappings without non-wal£*