Blender-producing mechanisms and a dichotomy for local dynamics for heterodimensional cycles

Abstract

Blenders are special hyperbolic sets used to produce various robust dynamical phenomena which appear fragile at first glance. We prove for $C^r$ diffeomorphisms ($r=2,\dots,\infty,\omega$) that blenders naturally exist (without perturbation) near non-degenerate heterodimensional cycles of coindex-1, and the existence is determined by arithmetic properties of moduli of topological conjugacy for diffeomorphisms with heterodimensional cycles. In particular, we obtain a $C^r$-generic dichotomy for dynamics in any small neighborhood $U$ of a non-degenerate heterodimensional cycle: either there exist infinitely many blenders accumulating on the cycle, forming robust heterodimensional dynamics in most cases, or there are no orbits other than those constituting the cycle lying entirely in $U$.