Joining rigidity for rational maps

We initiate a joining rigidity theory for rational maps on the Riemann sphere $\mathbb P^1=\mathbb P^1(\mathbb C)$. Let $f_1,f_2\colon\mathbb P^1\to\mathbb P^1$ be rational maps of degree at least $2$, and $μ_1,μ_2$ their respective measures of maximal entropy, whose supports are the Julia sets $J(f