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 $f_1,f_2\text{\hspace{0.17em}⁣}:{\mathbb{P}}^1\to {\mathbb{P}}^1$ be rational maps of degree at least $2$, and ${\mu }_1,{\mu }_2$ their respective measures of maximal entropy, whose supports are the Julia sets $J(f_1)$ and $J(f_2)$. We study ergodic joinings of the systems $(J(f_1),f_1,{\mu }_1)$ and $(J(f_2),f_2,{\mu }_2)$, namely ergodic probability measures on $J(f_1)\times J(f_2)$ which are invariant under $f_1\times f_2$ and whose marginals are ${\mu }_1$ and ${\mu }_2$. Our main theorem shows that a positive-mass local holomorphic relation forces algebraic rigidity. More precisely, if the joining charges the graph of a local biholomorphism, then that local relation globalizes to an invariant algebraic curve and yields either a finite cycle of rational graph or transpose-graph relations, or a genuinely multi-valued invariant algebraic correspondence. If no local biholomorphic graph has positive joining measure, then the joining generates a compact non-discrete family of local holomorphic relations. The proof introduces normalized inverse branch transfer maps and studies their cluster limits. Starting from a local biholomorphic graph of positive joining measure, recurrence and contraction of inverse branches produce recurrent local intertwining relations. These are promoted to an algebraic relation by a local-to-global rigidity argument in the non-Lattès case and by affine uniformization in the Lattès case. In the absence of any positive-mass local biholomorphic graph, the cluster family must be infinite, and its non-discrete closure gives the second alternative.