It is known that the famous conjecture that the Mandelbrot set is locally connected is equivalent to the combinatorial rigidity conjecture for the quadratic family.
Henriksen showed the existence of non-combinatorially rigid cubic polynomials, which have infinitely many capture renormalizations.
In this talk, we show that unicritical cubic polynomials can be non-combinatorially rigid in the whole cubic family by showing the existence of cubic polynomials with two distinct critical points having infinitely many cubic renormalizations.
We also discuss their dynamical properties using near-parabolic renormalization.
