The main cubioid and the modulus of renormalization

The main cubioid (CU) is a central part in the parameter space of cubic polynomials of one complex variable viewed as dynamical systems. It plays a similar role to that of the main cardioid in the (quadratic) Mandelbrot set, hence the title. However, in contrast to the main cardioid, topology and combinatorics of the CU are rather involved. We discuss a dynamical characterization of the CU, a proof of which has been recently completed. The last (most recent) ingredient is an upper bound on the moduli of quadratic-like restrictions. This is a joint project with A. Blokh and L. Oversteegen.