Coexistence of non-periodic attractors

TL;DR

This paper demonstrates the existence of codimension 3 laminations of polynomial maps in R^2 with multiple period doubling Cantor attractors, revealing complex coexistence phenomena in dynamical systems.

Contribution

It introduces new codimension 3 laminations of maps with multiple attractors and describes their geometric and topological properties within polynomial maps of degree at least two.

Findings

Existence of codimension 3 laminations with multiple attractors

Laminations are real-analytic with uniform diameter

Laminations' leaves align with eigenvalue foliation asymptotically

Abstract

In the space of polynomial maps of $\mathbb R^2$ of degree at least two, there are codimension $3$ laminations of maps with at least $3$ period doubling Cantor attractors. The leafs of the laminations are real-analytic and they have uniform diameter. The closure of each lamination contains the codimension one tangency locus of a saddle point. Asymptotically, the leafs of each lamination align with the leafs of the eigenvalue foliation. This is an example of general coexistence theorems valid for higher dimensional real-analytic unfoldings of two dimensional homoclinic tangencies.