One-dimensional Functional Equations

Abstract

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.