We discuss the (heterodimensional) cycles stabilization problem. For that we introduce robust cycles and explain how the stabilization of a cycle depends on the degree of differentiability considered. We close the first part by summarizing some results in the $C^1$-topology which depend on the notion of a blender horseshoe.
The next step is to discuss the stabilization problem in higher differentiability settings focusing on the so-called heterodimensional tangencies. The main steps is to see that center-unstable Hénon-like families yield blender horseshoes and that these families appear as limit families of some heterodimensional cycles with heterodimensional tangencies. Finally, we present a setting where the stabilization of cycles is obtained. The general principle is that we can obtain $C^2$-stabilization of cycles by adding geometrical complexity to the cycles.
In this talk we will explain all ingredients considered: heterodimensional cycles, heterodimensional tangencies, robust cycles, blender horseshoes, center-unstable Hénon-like families, and renormalization.