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Infinite chains of perfect fits for expanding Thurston maps

  • Ino Loukidou
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Abstract

The topological mating of two postcritically finite polynomials with dendritic Julia sets is encoded by a pair of circle laminations Λ±, whose collapse produces a sphere-filling curve. When a leaf of Λ+ and a leaf of Λ− share an endpoint they form a perfect fit. An unpublished proposition of Epstein, recorded by Petersen and Meyer, shows that for matings of honest degree-d polynomials an infinite-diameter ray equivalence class, i.e. an infinite chain of perfect fits is impossible. We show that this finiteness is a genuinely holomorphic phenomenon. Allowing the dynamics to carry a periodic critical orbit, we construct combinatorially expanding Thurston maps-realized by no expanding rational map-that admit invariant sphere-filling curves yet whose laminations Λ± contain infinite chains of perfect fits, in fact infinitely many of them.

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Cite

BibTeXRIS
BibTeX
@misc{loukidou2026infinite,
  title = {{Infinite chains of perfect fits for expanding Thurston maps}},
  author = {Ino Loukidou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03838},
  eprint = {2609.03838},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03838},
}
RIS
TY  - UNPB
TI  - Infinite chains of perfect fits for expanding Thurston maps
AU  - Ino Loukidou
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03838
UR  - https://arxiv.org/abs/2609.03838
AB  - The topological mating of two postcritically finite polynomials with dendritic Julia sets is encoded by a pair of circle laminations Λ^±, whose collapse produces a sphere-filling curve. When a leaf of Λ^+ and a leaf of Λ^− share an endpoint they form a perfect fit. An unpublished proposition of Epstein, recorded by Petersen and Meyer, shows that for matings of honest degree-d polynomials an infinite-diameter ray equivalence class, i.e. an infinite chain of perfect fits is impossible. We show that this finiteness is a genuinely holomorphic phenomenon. Allowing the dynamics to carry a periodic critical orbit, we construct combinatorially expanding Thurston maps-realized by no expanding rational map-that admit invariant sphere-filling curves yet whose laminations Λ^± contain infinite chains of perfect fits, in fact infinitely many of them.
ER  -