# Topological classification of limit periodic sets of polynomial planar vector fields

## TL;DR

This paper characterizes the topological structure of limit periodic sets in polynomial planar vector fields, showing they are equivalent to certain semialgebraic sets on the sphere, and vice versa.

## Contribution

It provides a complete topological classification of limit periodic sets in polynomial planar vector fields, establishing a correspondence with semialgebraic sets on the sphere.

## Key findings

- Limit periodic sets are topologically equivalent to compact, connected semialgebraic sets with empty interior on the sphere.
- Any such semialgebraic set can be realized as a limit periodic set in polynomial planar vector fields.
- The characterization is up to homeomorphisms, providing a topological classification.

## Abstract

We characterize the limit periodic sets of families of polynomial planar vector fields up to homeomorphisms. We show that any limit periodic set is topologically equivalent to a compact and connected semialgebraic set of the sphere with empty interior. Conversely, we show that any compact and connected semialgebraic set of the sphere with empty interior can be realized as a limit periodic set.