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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. 

Abstract We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows. 

The moduli space M ¯ 0 , n \overline {\mathcal {M}}_{0,n} carries a codimension- d d Chow class κ d \kappa _{d} . We consider the subspace K n d \mathcal {K}^{d}_{n} of A d ( M ¯ 0 , n , Q ) A^d(\overline {\mathcal {M}}_{0,n},\mathbb {Q}) spanned by pullbacks of κ d \kappa _d via forgetful maps. We find a permutation basis for K n d \mathcal {K}^{d}_{n} , and describe its annihilator under the intersection pairing in terms of d d -dimensional boundary strata. As an application, we give a new permutation basis of the divisor class group of M ¯ 0 , n \overline {\mathcal {M}}_{0,n} . 

We focus on various dynamical invariants associated to monomial correspondences on toric varieties, using algebraic and arithmetic geometry. We find a formula for their dynamical degrees, relate the exponential growth of the degree sequences to a strict log-concavity condition on the dynamical degrees and compute the asymptotic rate of the growth of heights of points of such correspondences. 

Let ϕ : S 2 → S 2 \phi :S^2 \to S^2 be an orientation-preserving branched covering whose post-critical set has finite cardinality n n . If ϕ \phi has a fully ramified periodic point p ∞ p_{\infty } and satisfies certain additional conditions, then, by work of Koch, ϕ \phi induces a meromorphic self-map R ϕ R_{\phi } on the moduli space M 0 , n \mathcal {M}_{0,n} ; R ϕ R_{\phi } descends from Thurston’s pullback map on Teichmüller space. Here, we relate the dynamics of R ϕ R_{\phi } on M 0 , n \mathcal {M}_{0,n} to the dynamics of ϕ \phi on S 2 S^2 . Let ℓ \ell be the length of the periodic cycle in which the fully ramified point p ∞ p_{\infty } lies; we show that R ϕ R_{\phi } is algebraically stable on the heavy-light Hassett space corresponding to ℓ \ell heavy marked points and ( n − ℓ ) (n-\ell ) light points. 

Let $$\unicode[STIX]{x1D719}$$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $$\unicode[STIX]{x1D719}$$ on Teichmüller space descends to a multivalued self-map—a Hurwitz correspondence $${\mathcal{H}}_{\unicode[STIX]{x1D719}}$$ —of the moduli space $${\mathcal{M}}_{0,\mathbf{P}}$$ . We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees . We show that the sequence of dynamical degrees of $${\mathcal{H}}_{\unicode[STIX]{x1D719}}$$ is always non-increasing and that the behavior of this sequence is constrained by the behavior of $$\unicode[STIX]{x1D719}$$ at and near points of its post-critical set.