On Surface Attractors and Repellers in 3-Manifolds

Abstract

We show that if f : M 3 → M 3 is an A-diffeomorphism with a surface two-dimensional attractor or repeller B and M 2 B is a supporting surface for B, then B = M 2 B and there is k ≥ 1 such that:i is homeomorphic to the 2-torus T 2 . 2) the restriction of f k to M 2 i (i ∈ {1, . . . , k}) is conjugate to Anosov automorphism of T 2 .

The topological classification of structurally stable 3-diffeomorphisms with two-dimensional basic sets

“…It follows from [11] and [18] that any manifold M 3 which admits structurally stable diffeomorphism → f M M : 3 3 with a two-dimensional expanding attractor (contracting repeller), is diffeomorphic to the torus T 3 and, moreover, f is topologically conjugated with the diffeomorphism obtained from the Anosov diffeomorphism by the generalized surgery operation. According to [9] each connected component of the two-dimensional surface basic set for A-diffeomorphism → f M M : 3 3 is homeomorphic to the torus and a restriction of some degree of f to this component is topologically conjugate with the Anosov diffeomorphism.…”

“…A contracting repeller of a diffeomorphism f is an expanding attractor of f −1 . According to [7], a basic set B of a diffeomorphism…”

“…The surface B M 2 is called the support of B. The following statement on surface basic sets follows from [9]. 3 3 the following holds:…”

“…B has type (2, 1) ((1, 2)) and, therefore, is not an expanding attractor (an contracting repeller). B coincides with its support and is a finite union of manifolds tamely embedded 6 in M 3 and homeomorphic to the 2-torus 7 . the restriction of B f k to any connected component of the support is conjugated to some hyperbolic automorphism of the torus.…”

“…Otherwise the embedding λ is said to be wild and the manifold X is said to be wildly embedded. 7 It should be emphasized that the support of a two-dimensional surface basic set is not necessarily smooth (the corresponding example is given in [15]). Due to [19], 2-torus B is tamely embedded in M 3 if and only if there is a topological embedding…”