Singularity theory and equivariant symplectic maps
This Lecture Notes Volume studies singularities of symplectic maps on \(\mathbb{R}^{2n}\). It consists of three parts: 1. A general theory for bifurcating period-\(q\) points of equivariant symplectic maps. 2. An introduction to a singularity theory framework for equivariant gradient bifurcation problems used in classification of singularities of bifurcating period-\(q\) points of symplectic maps. 3. A compendium of basic questions and results for symplectic maps on \(\mathbb{R}^{2n}\) and their generating functions. From the contents: After the introduction, as a fundamental tool Lagrangian generating functions are introduced in Chapter 2. Chapters 3 to 6 cover a singularity theory for equivariant gradient bifurcation problems. The general theory is introduced in Chapter 3, and in Chapter 4 it is applied to Lagrangian generating functions of \(\mathbb{Z}_ 2\) -- equivariant maps on \(\mathbb{R}^ 2\) -- with distinguished parameter. In Chapter 6 a classification in \(\mathbb{R}^ 2\) is given. Chapters 5 and 7 consider 2-parameter area preserving maps. Chapter 8 treats the effect of spatial symmetries on bifurcating period \(q\)-points and Chapter 9 the collision of multipliers at rational points. In the last chapter equivariant symplectic maps with collision of multipliers in \(\mathbb{R}^ 4\) are studied. The volume is mainly a research monograph, however an extensive appendix covers many basic results on symplectic maps for \(\mathbb{R}^{2n}\) and their generating functions.
scientific article; zbMATH DE number 482865