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1. Geodesic planes in geometrically finite acylindrical -manifolds

   https://doi.org/10.1017/etds.2021.19  

   BENOIST, YVES; OH, HEE (February 2022, Ergodic Theory and Dynamical Systems)

   Abstract Let M be a geometrically finite acylindrical hyperbolic $$3$$ -manifold and let $M^*$ denote the interior of the convex core of M . We show that any geodesic plane in $M^*$ is either closed or dense, and that there are only countably many closed geodesic planes in $M^*$ . These results were obtained by McMullen, Mohammadi and Oh [Geodesic planes in hyperbolic 3-manifolds. Invent. Math. 209 (2017), 425–461; Geodesic planes in the convex core of an acylindrical 3-manifold. Duke Math. J. , to appear, Preprint , 2018, arXiv:1802.03853] when M is convex cocompact. As a corollary, we obtain that when M covers an arithmetic hyperbolic $$3$$ -manifold $$M_0$$ , the topological behavior of a geodesic plane in $M^*$ is governed by that of the corresponding plane in $$M_0$$ . We construct a counterexample of this phenomenon when $$M_0$$ is non-arithmetic. 

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2. Simple weight modules with finite weight multiplicities over the Lie algebra of polynomial vector fields

   https://doi.org/10.1515/crelle-2022-0053  

   Grantcharov, Dimitar; Serganova, Vera (September 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract Let 𝒲 n {{\mathcal{W}}_{n}} be the Lie algebra of polynomial vector fields.We classify simple weight 𝒲 n {{\mathcal{W}}_{n}} -modules M with finite weight multiplicities. We prove that every such nontrivial module M is either a tensor module or the unique simple submodule in a tensor module associatedwith the de Rham complex on ℂ n {\mathbb{C}^{n}} . 

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3. THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

   https://doi.org/10.1017/S1474748019000318  

   Corvaja, Pietro; Ghioca, Dragos; Scanlon, Thomas; Zannier, Umberto (March 2021, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract Let $$K$$ be an algebraically closed field of prime characteristic $$p$$ , let $$X$$ be a semiabelian variety defined over a finite subfield of $$K$$ , let $$\unicode[STIX]{x1D6F7}:X\longrightarrow X$$ be a regular self-map defined over $$K$$ , let $$V\subset X$$ be a subvariety defined over $$K$$ , and let $$\unicode[STIX]{x1D6FC}\in X(K)$$ . The dynamical Mordell–Lang conjecture in characteristic $$p$$ predicts that the set $$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$$ is a union of finitely many arithmetic progressions, along with finitely many $$p$$ -sets, which are sets of the form $$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$$ for some $$m\in \mathbb{N}$$ , some rational numbers $$c_{i}$$ and some non-negative integers $$k_{i}$$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $$X$$ is an algebraic torus, we can prove the conjecture in two cases: either when $$\dim (V)\leqslant 2$$ , or when no iterate of $$\unicode[STIX]{x1D6F7}$$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $$X$$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction. 

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4. Algebraic stability of meromorphic maps descended from Thurston’s pullback maps

   https://doi.org/10.1090/tran/8221  

   Ramadas, Rohini (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   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. 

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5. Sharp Finiteness Principles For Lipschitz Selections

   Charles Fefferman, Pavel Shvartsman (September 2018, Geometric and functional analysis)

   Let (M,ρ) be a metric space and let Y be a Banach space. Given a positive integer m, let F be a set-valued mapping from M into the family of all compact convex subsets of Y of dimension at most m. In this paper we prove a finiteness principle for the existence of a Lipschitz selection of F with the sharp value of the finiteness constant. 

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