Robust heterodimensional cycles of co-index two via split blending machines

TL;DR

This paper introduces split blending machines to generate and control robust heterodimensional cycles of co-index two in partially hyperbolic diffeomorphisms, expanding the understanding of cycle robustness and configurations.

Contribution

The paper extends Asaoka's blending machines to a partially hyperbolic setting, enabling the creation of robust heterodimensional cycles of co-indices one and two.

Findings

Existence of non-escaping cycles involving strong stable and unstable manifolds.

Approximation of diffeomorphisms with robust heterodimensional cycles of co-indices one and two.

Application to skew product dynamics and matrix cocycles on GL(3,R).

Abstract

We consider diffeomorphisms $f$ with heterodimensional cycles of co-index two, associated with saddles $P$ and $Q$ having unstable indices $\ell$ and $\ell+2$, respectively. In a partially hyperbolic setting, where a two-dimensional center direction and strong invariant manifolds are defined, we introduce the class of \emph{non-escaping cycles}, where the strong stable manifold of $P$ and the strong unstable manifold of $Q$ are involved in the cycle. This configuration guarantees the existence of orbits that remain in a neighbourhood of the cycle. We show that such diffeomorphisms $f$ can be $C^1$ approximated by diffeomorphisms exhibiting simultaneously $C^1$ robust heterodimensional cycles of co-indices one and two, encompassing all possible combinations among hyperbolic sets of unstable indices $\ell$, $\ell+1$, and $\ell+2$. The proof relies on the construction of \emph{split blending machines}. This tool extends Asaoka's blending machines to a partially hyperbolic setting, providing a mechanisms to generate and control robust intersections within a two-dimensional central bundle. We also present simple dynamical settings where such cycles occur, namely skew product dynamics with surface fiber maps. Non-escaping cycles also appear in contexts such as Derived from Anosov diffeomorphisms and matrix cocycles on $\mathrm{GL}(3,\mathbb{R})$.