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Abstract We establish an implication between two long-standing open problems in complex dynamics. The roots of the $$n$$th Gleason polynomial $$G_{n}\in{\mathbb{Q}}[c]$$ comprise the $$0$$-dimensional moduli space of quadratic polynomials with an $$n$$-periodic critical point. $$\operatorname{Per}_{n}(0)$$ is the $$1$$-dimensional moduli space of quadratic rational maps on $${\mathbb{P}}^{1}$$ with an $$n$$-periodic critical point. We show that if $$G_{n}$$ is irreducible over $${\mathbb{Q}}$$, then $$\operatorname{Per}_{n}(0)$$ is irreducible over $${\mathbb{C}}$$. To do this, we exhibit a $${\mathbb{Q}}$$-rational smooth point on a projective completion of $$\operatorname{Per}_{n}(0)$$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $$n$$, $$\operatorname{Per}_{n}(0)$$ itself has no $${\mathbb{Q}}$$-rational points.
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End‐periodic homeomorphisms and volumes of mapping tori
Abstract Given an irreducible, end‐periodic homeomorphism of a surface with finitely many ends, all accumulated by genus, the mapping torus, , is the interior of a compact, irreducible, atoroidal 3‐manifold with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of in terms of the translation length of on the pants graph of . This builds on work of Brock and Agol in the finite‐type setting. We also construct a broad class of examples of irreducible, end‐periodic homeomorphisms and use them to show that our bound is asymptotically sharp.
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- Award ID(s):
- 1902729
- PAR ID:
- 10420594
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of Topology
- Volume:
- 16
- Issue:
- 1
- ISSN:
- 1753-8416
- Page Range / eLocation ID:
- p. 57-105
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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