2025/11/16 by Pablo G. Barrientos, Lorenzo J. Díaz, Barrientos, Pablo G. +7
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2511.12412
openalex publication_date 2025/11/16 · openalex created_date 2025/11/19 · openalex updated_date 2026/07/28
We consider diffeomorphisms f with heterodimensional cycles of co-index two, associated with saddles P and Q having unstable indices ℓ and ℓ+2, respectively. In a partially hyperbolic setting, where a two-dimensional center direction and strong invariant manifolds are defined, we introduce the class of 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 C1 approximated by diffeomorphisms exhibiting simultaneously C1 robust heterodimensional cycles of co-indices one and two, encompassing all possible combinations among hyperbolic sets of unstable indices ℓ, ℓ+1, and ℓ+2. The proof relies on the construction of 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 GL(3,ℝ).