发表机构
Università di Pisa; University of Oklahoma（比萨大学; 俄克拉荷马大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立有理映射的联结刚性理论：若最大熵测度的联结支持局部双全纯图，则其整体化为代数曲线，否则产生非离散局部全纯关系族。
AI 中文摘要
我们启动了黎曼球面 $\mathbb P^1=\mathbb P^1(\mathbb C)$ 上有理映射的联结刚性理论。设 $f_1,f_2\colon\mathbb P^1\to\mathbb P^1$ 为次数至少为 $2$ 的有理映射，$\mu_1,\mu_2$ 分别为它们的最大熵测度，其支集为 Julia 集 $J(f_1)$ 和 $J(f_2)$。我们研究系统 $(J(f_1),f_1,\mu_1)$ 与 $(J(f_2),f_2,\mu_2)$ 的遍历联结，即在 $J(f_1)\times J(f_2)$ 上在 $f_1\times f_2$ 作用下不变且边缘分布为 $\mu_1$ 和 $\mu_2$ 的遍历概率测度。我们的主定理表明，一个具有正质量的局部全纯关系迫使代数刚性成立。更精确地说，如果联结赋予某个局部双全纯映射的图以正质量，那么该局部关系可整体化为一条不变代数曲线，并产生有限个有理图或转置图关系的循环，或一个真正多值的不变代数对应。若没有局部双全纯图具有正联结测度，则联结生成一个紧的非离散局部全纯关系族。证明引入了归一化逆分支传递映射并研究其簇极限。从具有正联结测度的局部双全纯图出发，逆分支的回复性和收缩产生回复的局部交织关系。在非 Lattès 情形下，通过局部到整体的刚性论证，在 Lattès 情形下通过仿射一致化，这些关系被提升为代数关系。在不存在任何正质量局部双全纯图的情况下，簇族必定是无限的，其非离散闭包给出第二种备选情形。
英文摘要
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 $J(f_1)$ and $J(f_2)$. We study ergodic joinings of the systems $(J(f_1),f_1,μ_1)$ and $(J(f_2),f_2,μ_2)$, 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 $μ_1$ and $μ_2$. 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.
