Persistence of Heterodimensional Cycles

Dongchen Li, Dmitry Turaev

Published 2021-05-08Version 1

A heterodimensional cycle is an invariant set of a dynamical system consisting of two hyperbolic periodic orbits with different dimensions of their unstable manifolds and a pair of orbits that connect them. For systems which are at least C^2, we show that bifurcations of a coindex-1 heterodimensional cycle within a generic 2-parameter family always create robust heterodimensional dynamics, i.e., chain-transitive sets which contain coexisting orbits with different numbers of positive Lyapunov exponents and persist for an open set of parameter values. In particular, we solve the so-called $C^r$-stabilization problem for the coindex-1 heterodimensional cycles in any regularity class $r=2,\ldots,\infty,\omega$. The results are based on the observation that arithmetic properties of moduli of topological conjugacy of systems with heterodimensional cycles determine the emergence of Bonatti-D\'iaz blenders.