Joining rigidity for rational maps
Abstract
We initiate a joining rigidity theory for rational maps on the Riemann sphere ℙ1=ℙ1(ℂ). Let f1,f2:ℙ1→ℙ1 be rational maps of degree at least $2$, and μ1,μ2 their respective measures of maximal entropy, whose supports are the Julia sets J(f1) and J(f2). We study ergodic joinings of the systems (J(f1),f1,μ1) and (J(f2),f2,μ2), namely ergodic probability measures on J(f1)× J(f2) which are invariant under f1× f2 and whose marginals are μ1 and μ2. 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.
Related
BibTeX
@misc{bianchi2026joining,
title = {{Joining rigidity for rational maps}},
author = {Fabrizio Bianchi and Yan Mary He},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14890},
eprint = {2609.14890},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14890},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Joining rigidity for rational maps
AU - Fabrizio Bianchi
AU - Yan Mary He
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14890
UR - https://arxiv.org/abs/2609.14890
AB - We initiate a joining rigidity theory for rational maps on the Riemann sphere ℙ^1=ℙ^1(ℂ). Let f_1,f_2:ℙ^1→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)× J(f_2) which are invariant under f_1× 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.
ER -