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-wandering points.- 2.6.2 C0-embedding.- 2.6.3 Diffeomorphisms with a unique fixed point.- 2.6.4 Diffeomorphisms with several fixed points.- 2.7 Comments and references.- 3 Generalized Abel Equation.- 3.1 Local solvability.- 3.1.1 Local solvability in a neighborhood of a non-fixed point.- 3.1.2 Proof of Theorem 3.1 for analytic functions.- 3.1.3 Local solvability at an isolated fixed point.- 3.1.4 More on analytic solutions.- 3.2 Global solutions of equations with not more than one fixed point.- 3.2.1 Equations with fixed-point free mappings F.- 3.2.2 The case of a single fixed point.- 3.3 Gluing method for linear equations with several fixed points.- 3.3.1 Cohomological equation.- 3.3.2 Equations with hyperbolic fixed points.- 3.4 Comments and references.- 4 Equations with Several Transformations of Argument.- 4.1 Local solvability.- 4.2 Extension of solutions.- 4.2.1 Absorbers.- 4.2.2 Extension of solutions from an absorber.- 4.2.3 Extension from intersection of absorbers. Decomposition method.- 4.3 Examples.- 4.4 Difference equations in Carleman classes.- 4.4.1 Decomposition in classes C(mn).- 4.4.2 Equations with constant coefficients.- 4.4.3 Equations with non-constant coefficients.- 4.5 Comments and references.- 5 Linear Equations.- 5.1 Generalized linear Abel equation.- 5.1.1 Equations on the real line with a unique fixed point.- 5.1.2 Cohomological equation.- 5.1.3 Spectrum of a weighted shift operator.- 5.1.4 Normal solvability of equations with hyperbolic fixed points.- 5.1.5 Equations with periodic points.- 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.
