Holomorphic Dynamics in Loop Spaces

Abstract

We discuss a certain type of evolution of loops in smooth manifolds described in terms of natural almost complex structures on appropriate loop spaces. The main attention is paid to holomorphic families of transverse loops in a Cauchy---Riemann manifold, with an emphasis on the geometry of their images. In this context, we introduce a special class of foliated solid tori, called holomorphic doughnuts, and show that they are connected with certain classical geometric constructions. In particular, we show that holomorphic doughnuts can be obtained from isolated singularities of algebraic plane curves. We also establish the existence of holomorphic dynamics for generic real-analytic loops.