Topological equivalence of homotopic vector fields

Question

Two (tangent) vector fields $X$ and $Y$ on oriented differentiable manifolds $M$ and $N$, respectively, are topologically equivalent, if there is an orientation-preserving homeomorphism $M \to N$, which sends orbits of $X$ to orbits of $Y$, preserving the direction of the orbits. The vector fields are topologically conjugate, if there is a homeomorphism $h: M \to N$, such that $\phi^X_t = h^{-1} \circ \phi^Y_t \circ h$, where $\phi^X$, $\phi^Y$ are the flows of $X$ and $Y$, respectively. A vector field is structurally stable, if small perturbations of it result in a topologically equivalent vector field.

In dynamical systems, structural stability is an important and well-studied aspect, but one is also interested in comparing the qualitative behavior of vector fields, which are not just small perturbations from each other. In order to show the equivalence or conjugacy of vector fields in concrete cases, one can sometimes construct the homeomorphism by hand. Instead of showing that two vector fields are topologically equivalent, it is often much easier to show, that they are homotopic (as sections of the tangent bundle) via vector fields which preserve a property. In concrete examples, one might like to know, what happens to the qualitative behavior of a dynamical system, if one varies parameters. Therefore it would be nice to have a few general results that under certain conditions homotopic vector fields are topologically equivalent. I am surprised that I could not find anything in the literature addressing this.

Homotopy classes of non-singular vector fields have been studied a lot in the literature, but in dynamical systems singular vector fields play an important role. Since all vector fields are homotopic, only homotopies, which preserve certain properties of the vector fields, have the chance of giving topologically equivalent vector fields. Most importantly from a dynamical point of view, the singular sets should be preserved by the homotopy up to homeomorphism.

For example, a homotopy along structurally stable vector fields gives topologically equivalent vector fields at the endpoints. However, it might be difficult to confirm structural stability for all vector fields along the homotopy. Also, there should be weaker conditions under which homotopic vector fields are topologically equivalent. There is a theorem by Shub proving topological conjugacy for homotopic expanding endomorphisms on a compact manifold, which could be relevant, but I haven't found statement/proof in the vector field setting.

Does anybody know a reference for Shub's theorem in the vector field setting? Does anybody know results about when certain homotopic vector fields are topologically equivalent? If the local behaviour of the homotopy around the singularities is known, are there methods to deduce something about the global behaviour?

Answer 1

I don't know much about the general setting you discuss, but a reasonable notion of homotopy between singular vector fields in the local holomorphic setting has been given, and studied, by J.-F. Mattei in a French article

Modules de feuilletages holomorphes singuliers. I. Équisingularité

The idea is to require that the 1-dimensional foliation induced by the integral curves of the vector fields $X$ and $Y$ be embedded in an integrable 2-dimensional foliation (a kind of foliated cobordism) with special assumptions about the deformation of the singularities ("equisingularity"). $X$ and $Y$ will then be topologically conjugate as germs of a vector field. As for having global results, even in the complex analytic realm, I don't know of general theorems. You'll probably need to impose rather restrictive conditions on the deformation of the local type of the singularities, making it barely usable (though it is only a guess). The point is that the generic holomorphic foliation on a compact surface is rather rigid, meaning that topological conjugacy often leads to analytical conjugacy. This is particularly the case for complex foliations of the complex projective plane, as was proved by Y. Il'Yashenko

Topology of phase portraits of analytic differential equations on a complex projective plane

(I'm afraid this link only points a Russian text, yet a translation of this paper exists somewhere, e.g. on some shelf in my out-of-reach-for-now office...)