Invariant escaping Fatou components with two rank-one limit functions for automorphisms of

We construct automorphisms of ${\mathbb C}^2$ , and more precisely transcendental Hénon maps, with an invariant escaping Fatou component which has exactly two distinct limit functions, both of (generic) rank one. We also prove a general growth lemma for the norm of points in orbits belonging to invariant escaping Fatou components for automorphisms of the form $F(z,w)=(g(z,w),z)$ with $g(z,w):{\mathbb C}^2\rightarrow {\mathbb C}$ holomorphic.

Generic rank of Betti map and unlikely intersections

Let $\mathcal {A} \rightarrow S$ be an abelian scheme over an irreducible variety over $\mathbb {C}$ of relative dimension $g$. For any simply-connected subset $\Delta$ of $S^{\mathrm {an}}$ one can define the Betti map from $\mathcal {A}_{\Delta }$ to $\mathbb {T}^{2g}$, the real torus of dimension $2g$, by identifying each closed fiber of $\mathcal {A}_{\Delta } \rightarrow \Delta$ with $\mathbb {T}^{2g}$ via the Betti homology. Computing the generic rank of the Betti map restricted to a subvariety $X$ of $\mathcal {A}$ is useful to study Diophantine problems, e.g. proving the geometric Bogomolov conjecture over char $0$ and studying the relative Manin–Mumford conjecture. In this paper we give a geometric criterion to detect this rank. As an application we show that it is maximal after taking a large fibered power (if $X$ satisfies some conditions); it is an important step to prove the bound for the number of rational points on curves (Dimitrov et al., Uniformity in Mordell–Lang for Curves, Preprint (2020), arXiv:2001.10276). Another application is to answer a question of André, Corvaja and Zannier and improve a result of Voisin. We also systematically study its link with the relative Manin–Mumford conjecture, reducing the latter to a simpler conjecture. Our tools are functional transcendence and unlikely intersections for mixed Shimura varieties.

Mixed Ax–Schanuel for the universal abelian varieties and some applications

In this paper we prove the mixed Ax–Schanuel theorem for the universal abelian varieties (more generally any mixed Shimura variety of Kuga type), and give some simple applications. In particular, we present an application for studying the generic rank of the Betti map.

Tensor decompositions and rank increment conjecture

Some properties of tensor ranks and the non-closeness issue of sets with given restrictions on the rank of tensors entering those sets are studied. It is proved that the rank of the d-dimensional Laplacian equals d. The following conjecture is formulated: for any tensor of non-maximal rank there exists a nonzero decomposable tensor (tensor of rank 1) such that the rank increases by one after adding this tensor. In the general case, it is proved that this property holds algebraically almost everywhere for complex tensors of fixed size whose rank is strictly less than the generic rank.

On the status of Kaiparaites Matsumoto, 1955 (Cretaceous; Ammonoidea: Kossmaticeratidae)

The kossmaticeratid ammonite Kaiparaites was on its inception synonymized by its author (Matsumoto 1955) with Natalites (Collignon 1954). However, close comparison of the type species of these two genera shows them to be morphologically and stratigraphically distinct. Whereas Santonian Natalites shares the flexed constrictions of late Turonian–Coniacian Kossmaticeras, the straight constrictions of late Campanian – Maastrichtian Kaiparaites are shared with Caledonites, Gunnarites, Grossouvrites and Jacobites. Consequently, Kaiparaites is resurrected from the synonymy of Natalites and demonstrates that, under phylogenetic consideration, characters first thought to be of prime taxonomic significance actually turn out to be secondary. Since treatment of Natalites as a subgenus of Kossmaticeras emphasizes primitive characters, it is returned to generic rank, as is Santonian Karapadites although, for those who find use for subgenera, the latter can be treated as a subgenus of Natalites.

New and little-known taxa of the tribe Diestramimini (Orthoptera: Rhaphidophoridae: Aemodogryllinae) from Southeast Asia. Part 2

The genera Megadiestramima Stor. et Gor. and Mimadiestra Stor. et Dawwrueng are considered. Megadiestramima is here divided into three subgenera: Megadiestramima s. str.; Leodiestramima Stor. (in 2014, this subgenus was raised up to generic rank by Storozhenko & Dawwrueng, but in 2015, it was restored in original subgeneric rank by Gorochov & Storozhenko); Neodiestramima subgen. nov. Seven new species and subspecies of this genus are described from Vietnam, Thailand and Cambodia: M. (M.) borealis sp. nov.; M. (M.) abramovi sp. nov.; M. (M.) bilobata sp. nov.; M. (M.) centralis sp. nov.; M. (N.) orlovi lata subsp. nov.; M. (N.) o. khmerica subsp. nov.; M. (N.) brevispina sp. nov. Keys to subgenera and species of Megadiestramima s. l. and Mimadiestra as well as keys to subgenera of Diestramima Stor. and Adiestramima Gor. are given. Eight additional new taxa of the latter genera and of the genus Tamdaotettix Gor. are described from Laos, China and Vietnam: Baculitettix subgen. nov. and Excisotettix subgen. nov. in Diestramima s. l.; D. (B.) propria apicalis subsp. nov.; Hamatotettix subgen. nov. and Ulterotettix subgen. nov. in Adiestramima s. l.; A. (Adiestramima) originalis sp. nov.; T. (Tamdaotettix) ailaoshanicus sp. nov.; T. (T.) minipullus sp. nov. New data on distribution of some species are also given.

A new combination in Dolabrifolia (Angraecinae, Orchidaceae)

Species in Angraecum sect. Dolabrifolia (Pfitzer 1889: 216) Garay (1973: 499) were assigned to several genera until Garay (1973) finally transferred the type species, A. distichum (Lindley 1836: t. 1781), along with A. podochiloides Schlechter (1906: 162) and A. aporoides Summerhayes (1964: 560) to Angraecum Bory (1804: 359) (see Simo-Droissart et al. 2016a). Later, Angraecum bancoense Burg in Arends et al. (1980: 26) and A. poppendickianum Szlachetko & Olszewski (2001: 884) were described and included in this section. According to Garay (1973), Angraecum sect. Dolabrifolia is characterized by very short, fleshy, laterally compressed and densely imbricate leaves, with a groove on the upper surface; the lateral compression being a unique feature within the genus. Based on this unique morphological character, Szlachetko and Romowicz (2007) raised the section to the rank of genus, Dolabrifolia (Pfitz.) Szlachetko & Romowicz (2007: 54) and proposed five new combinations, namely Dolabrifolia disticha (Lindl.) Szlachetko & Romowicz (2007: 54), D. aporoides (Summerh.) Szlachetko & Romowicz (2007: 54), D. bancoensis (Burg) Szlachetko & Romowicz (2007: 54), D. podochiloides (Schltr.) Szlachetko & Romowicz (2007: 54) and D. poppendickiana (Szlach. & Olszewski) Szlachetko & Romowicz (2007: 54). The taxonomic decision of Szlachetko and Romowicz (2007) was followed by Szlachetko et al. (2013), who also raised nine other Angraecum sections to the generic rank.