| Popis: |
We will study in this paper diffeomorphisms f : S3 → S3 with two invariant linked solenoids Σ+, Σ−, one attracting and one repelling. We will call such mappings linked solenoid mappings. Such mappings arise when studying Henon mappings in C2, and have been studied in [HO1], [HPV], [BS], and [Bu]. More precisely, a linked solenoid mapping is one for which the 3-sphere S3 can be cut into two linked unknotted solid tori T+, T−, such that f : T+ → T+ and f−1 : T−1 → T−1 are conjugate to the standard maps (see below) giving rise to solenoids, as first studied in [vD] and [V]. These maps are structurally stable, and hence our linked solenoid maps will be structurally stable on their non-wandering sets. But they are not structurally stable. In Section 3 of this paper we will define a conjugacy invariant ntl(f ) ⊂ (R/2πZ)2 of linked solenoid mappings mapping. We will compute ntl(f ) for some particular linked solenoid maps in Section 4, we will show it can take on an infinite-dimensional set of values in Section 5, and in Section 6 we will show that in an open set of solenoidal mappings it is a complete invariant and classifies these mappings up to topological conjugacy. There are many things about this invariant which we don’t know; we present some of these in Section 7. We thank Thierry Bousch, whose criticism of [HO1] prompted part of this paper. |
