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Register a new userSelf-indexing energy function for Morse-Smale diffeomorphisms on 3-manifolds
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The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy function is equivalent to the existence of a Heegaard splitting of a special type with respect to the considered diffeomorphism.
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This note deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from cite{GrLaPo}, cite{GrLaPo1}, where gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well dynamics of diffeomorphism. The paper is devoted to finding conditions to the existence of such an energy function, that is, a function whose set of critical points coincides with the non-wandering set of the considered diffeomorphism. We show that the necessary and sufficient conditions to the existence of a dynamically ordered energy function reduces to the type of embedding of one-dimensional attractors and repellers of a given Morse-Smale diffeomorphism on a closed 3-manifold.
Denote by $DC(M)_0$ the identity component of the group of the compactly supported $C^r$ diffeomorphisms of a connected $C^infty$ manifold $M$. We show that if $dim(M)geq2$ and $r eq dim(M)+1$, then any homomorphism from $DC(M)_0$ to ${Diff}^1(R)$ or ${Diff}^1(S^1)$ is trivial.
We construct a Floer type boundary operator for generalised Morse-Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the $mathbb{Z}_2$ homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.
We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is analogous to the correspondence between bi-contact structures and projectively or conformally Anosov flows on three-manifolds found by Eliashberg--Thurston and by Mitsumatsu.
We develop the idea of self-indexing and the technology of gradient-like vector fields in the setting of Morse theory on a complex algebraic stratification. Our main result is the local existence, near a Morse critical point, of gradient-like vector fields satisfying certain ``stratified dimension bounds up to fuzz for the ascending and descending sets. As a global consequence of this, we derive the existence of self-indexing Morse functions.
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