Jordan mating is always possible for polynomials

Abstract

Suppose $f$ and $g$ are two post-critically finite polynomials of degree $d_1$ and $d_2$ respectively and suppose both of them have a finite super-attracting fixed point of degree $d_0$. We prove that one can always construct a rational map $R$ of degree $$D = d_1 + d_2 - d_0$$ by gluing $f$ and $g$ along the Jordan curve boundaries of the immediate super-attracting basins. The result can be used to construct many rational maps with interesting dynamics.