Analytic varieties as limit periodic sets

TL;DR

The paper demonstrates that for most radii, one can construct analytic families of vector fields whose limit periodic sets correspond to the zero set of a real-analytic function within a disk, with polynomial cases also addressed.

Contribution

It establishes the existence of analytic and polynomial families of vector fields with prescribed limit periodic sets defined by analytic functions.

Findings

Existence of analytic families with prescribed limit sets for almost all radii.

Polynomial families can realize the same limit periodic sets when the defining function is polynomial.

Results apply to a broad class of real-analytic functions and their zero sets.

Abstract

Let $f(x,y) \not\equiv 0$ be a real-analytic planar function. We show that, for almost every $R>0$ there exists an analytic 1-parameter family of vector fields $X_{\lambda}$ which has $\{f(x,y)=0\} \cap \bar{B_R((0,0))}$ as a limit periodic set. Furthermore, we show that if $f(x,y)$ is polynomial, then there exists a polynomial family with these properties.