Non-autonomous vector fields on $S^3$: simple dynamics and wild separatrices embedding

TL;DR

This paper constructs examples of non-autonomous vector fields on the 3-sphere with simple dynamics but complex topological features, including wild separatrix embeddings and special integral curves with exponential dichotomy.

Contribution

It introduces new examples of non-autonomous vector fields with wild topological structures on $S^3$, combining techniques from diffeomorphism theory and suspensions.

Findings

Existence of vector fields with wild embedded separatrices.

Construction of vector fields with special saddle integral curves.

Examples include periodic, almost periodic, and nonrecurrent vector fields.

Abstract

We construct new substantive examples of non-autonomous vector fields on 3-dimensional sphere having a simple dynamics but non-trivial topology. The construction is based on two ideas: the theory of diffeomorpisms with wild separatrix embedding (Pixton, Bonatti-Grines, etc.) and the construction of a non-autonomous suspension over a diffeomorpism (Lerman-Vainshtein). As a result, we get periodic, almost periodic or even nonrecurrent vector fields which have a finite number of special integral curves possessing exponential dichotomy on $\R$ such that among them there is one saddle integral curve (with an exponential dichotomy of the type (3,2)) having wildly embedded two-dimensional unstable separatrix and wildly embedded three-dimensional stable manifold. All other integral curves tend, as $t\to \pm \infty,$ to these special integral curves. Also we construct another vector fields having $k\ge 2$ special saddle integral curves with tamely embedded two-dimensional unstable separatrices forming mildly wild frames in the sense of Debrunner-Fox. In the case of periodic vector fields, corresponding specific integral curves are periodic with the period of the vector field, and they are almost periodic for the case of almost periodic vector fields.