Robust heterodimensional cycles of co-index two via split blending machines
Pablo G. Barrientos, Lorenzo J. Díaz, Yuri Ki, Cristina Lizana, Sebastián A. Pérez
TL;DR
The paper addresses the construction of robust heterodimensional cycles of co-index two in a partially hyperbolic setting by introducing non-escaping cycles that couple the strong stable and strong unstable directions. It develops split blending machines, a two-dimensional-centre extension of Asaoka’s blending machinery, to generate and control robust intersections in a two-dimensional central bundle and to realize robust co-index one and two cycles simultaneously. The approach hinges on quotient center dynamics, skew-product representations, and carefully adapted perturbations that preserve contour structures, culminating in a tower-map framework where preblending and split blending machines yield robust cycles and explicit examples. The results significantly extend the toolkit for producing robust nonhyperbolic dynamics and provide concrete dynamical settings, including skew-products and GL$(3,\mathbb{R})$-cocycles, where non-escaping cycles arise and can be stabilized under perturbations.
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 $\mathrm{GL}(3,\mathbb{R})$.