Construction of Smooth Source–Sink Arcs in the Space of Diffeomorphisms of a Two-Dimensional Sphere

Abstract

It is well known that the mapping class group of the two-dimensional sphere \({{\mathbb{S}}^{2}}\) is isomorphic to the group \({{\mathbb{Z}}_{2}} = \{ - 1, + 1\} \). At the same time, the class +1(–1) contains all orientation-preserving (orientation-reversing) diffeomorphisms and any two diffeomorphisms of the same class are diffeotopic, that is, they are connected by a smooth arc of diffeomorphisms. On the other hand, each class of maps contains structurally stable diffeomorphisms. It is obvious that in the general case, the arc connecting two diffeotopic structurally stable diffeomorphisms undergoes bifurcations that destroy structural stability. In this direction, it is particular interesting in the question of the existence of a connecting them stable arc – an arc pointwise conjugate to arcs in some of its neighborhood. In general, diffeotopic structurally stable diffeomorphisms of the 2-sphere are not connected by a stable arc. In this paper, the simplest structurally stable diffeomorphisms (source–sink diffeomorphisms) of the 2-sphere are considered. The non-wandering set of such diffeomorphisms consists of two hyperbolic points: the source and the sink. In this paper, the existence of an arc connecting two such orientation-preserving (orientation-reversing) diffeomorphisms and consisting entirely of source-sink diffeomorphisms is constructively proved.