Infinitely Many Attracting Periodic Circles in Higher Dimensions

TL;DR

This paper demonstrates that in higher-dimensional dynamical systems with specific heteroclinic cycles, there exist open sets where a residual subset exhibits infinitely many attracting periodic circles, extending known phenomena.

Contribution

It introduces a new result showing the abundance of infinitely many attracting periodic circles near certain heteroclinic cycle maps in higher dimensions.

Findings

Existence of open sets with residual subsets having infinitely many attracting circles.

Use of rescaling to the standard Hénon map in the proof.

Corrected Lyapunov coefficient formula on the Neimark-Sacker line.

Abstract

We study $C^r$ ($5 \le r \le \infty$) diffeomorphisms on closed manifolds of dimension at least three with a heteroclinic cycle between two hyperbolic periodic points. At each point, the unstable direction is one dimensional, and the stable and unstable eigenvalues closest to $1$ in modulus are real and simple. One heteroclinic connection is transverse and the other is non-transverse, and the product of those two eigenvalues is less than $1$ at one point and greater than $1$ at the other. Arbitrarily close to such a map, there are open sets in which a residual subset of diffeomorphisms has infinitely many attracting normally hyperbolic periodic circles. The proof uses a rescaling to the standard H\'enon map and a corrected formula for the Lyapunov coefficient on its Neimark-Sacker (Andronov-Hopf) line.