Stabilization of heterodimensional cycles

TL;DR

This paper proves that most heterodimensional cycles, except a specific twisted case with extra restrictions, can be stabilized through nearby diffeomorphisms, ensuring robust cycles involving saddles of different indices.

Contribution

It establishes a broad stabilization result for heterodimensional cycles with index difference one, excluding a special twisted case with additional geometric constraints.

Findings

Most heterodimensional cycles with index difference one can be stabilized.

The only exception is a specific class of twisted cycles with extra geometric restrictions.

Stabilization leads to robust cycles involving hyperbolic sets.

Abstract

We consider diffeomorphisms $f$ with heteroclinic cycles associated to saddles $P$ and $Q$ of different indices. We say that a cycle of this type can be stabilized if there are diffeomorphisms close to $f$ with a robust cycle associated to hyperbolic sets containing the continuations of $P$ and $Q$. We focus on the case where the indices of these two saddles differ by one. We prove that, excluding one particular case (so-called twisted cycles that additionally satisfy some geometrical restrictions), all such cycles can be stabilized.