Surface basic sets with wildly embedded supporting surfaces

The situation where a “nice” diffeomorphism f of a 3-manifold has a wildly embedded invariant surfaceM for which the restriction g = f| _M : M → M is “nice” is considered.     

Classification of one-dimensional expanding attractors

It is shown that, in any dimension d ≥ 3, there exist diffeomorphisms of compact d-manifolds with one-dimensional expanding attractors which are conjugate on these attractors but not conjugate on their neighborhoods.     

Gradient flows with wildly embedded closures of separatrices

We show that for any n ≥ 4 there exists an n-dimensional closed manifold M ⁿ on which one can define a Morse-Smale gradient flow f ^t with two nodes and two saddles such that the closure of the separatrix of some saddle of f ^t is a wildly embedded sphere of codimension 2. We also prove that the closures of separatrices of a flow with three equilibrium points are always embedded in a locally flat way.     

Saddle-type solenoidal basis sets

Mathematical Notes - An example of a diffeomorphism of the 3-sphere with positive topological entropy which has a one-dimensional solenoidal basis set with a two-dimensional unstable and a...     

On Surface Attractors and Repellers in 3-Manifolds

We show that if f: M ³ → M ³ is an A diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal{B}$ with support $M_\mathcal{B}^2$ , then $\mathcal{B} = M_\mathcal{B}^2$ and there exists a k ≥ 1 such that (1) $M_\mathcal{B}^2$ is the disjoint union M ₁ ² ? ? ? M _k ² of tame surfaces such that each surface M _i ² is homeomorphic to the 2-torus T ²; (2) the restriction of f ^k to M _i ² , i ∈ {1,..., k}, is conjugate to an Anosov diffeomorphism of the torus T ².     

Underlying Manifolds of High-Dimensional Morse–Smale Diffeomorphisms with Two Saddle Periodic Points

Mathematical Notes - The paper describes the topological structure of closed manifolds of dimension $$\ge4$$ that admit Morse–Smale diffeomorphisms whose nonwandering sets contain arbitrarily...     

Global dynamics of Morse-Smale systems

This paper is a survey of relatively recent results on the classification of Morse-Smale dynamical systems on closed manifolds. It also contains both old and relatively recent results on the relationship between the topology of the ambient manifold and the dynamical characteristics of Morse-Smale systems. Original Russian Text ? E.V. Zhuzhoma, V.S. Medvedev, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 261, pp. 115–139. Dedicated to V.Z Grines on the occasion of his 60th birthday