Fragile cycles

TL;DR

This paper investigates fragile heterodimensional cycles in diffeomorphisms, constructing a codimension one submanifold of such systems that lack robust cycles, applicable to manifolds of dimension four or higher.

Contribution

It introduces a construction of a codimension one submanifold of diffeomorphisms with fragile heterodimensional cycles in dimensions four and above.

Findings

Constructs a codimension one submanifold of diffeomorphisms with fragile cycles.

Shows the existence of fragile heterodimensional cycles in higher-dimensional manifolds.

Applicable to manifolds of dimension ≥ 4.

Abstract

We study diffeomorphisms $f$ with heterodimensional cycles, that is, heteroclinic cycles associated to saddles $p$ and $q$ with different indices. Such a cycle is called fragile if there is no diffeomorphism close to $f$ with a robust cycle associated to hyperbolic sets containing the continuations of $p$ and $q$. We construct a codimension one submanifold of $\Difr(\SS^2\times \SS^1)$, $r\ge 1$, that consists of diffeomorphisms with fragile heterodimensional cycles. Our construction holds for any manifold of dimension $\ge 4$.