High rank invariant subvarieties

Invariant Subvarieties of Minimal Homological Dimension, Zero Lyapunov Exponents, and Monodromy

We classify the GL(2,R)-invariant subvarieties M in strata of Abelian differentials for which any two M-parallel cylinders have homologous core curves. This answers a question of Mirzakhani and Wright. As a corollary we show that outside of an explicit list of exceptions, if M is a GL(2,R)-invariant subvariety, then the Kontsevich-Zorich cocycle has nonzero Lyapunov exponents in the symplectic orthogonal of the projection of the tangent bundle of M to absolute cohomology.

具有二次定义域的代数本原不变子簇

Algebraically primitive invariant subvarieties with quadratic field of definition

本文证明了具有二次定义域的平移曲面分层中，代数本原不变子簇仅包括十边形、Weierstrass曲线、亏格二特征形式轨迹及亏格四极小分层中的秩二示例。

$\mathbf P^n$ 的全不变子簇

Totally invariant subvarieties of $\mathbf P^n$

本文证明了在复射影空间上次数大于1的自同态下，满足原像等于自身的不可约Zariski闭子集必为线性子空间。

Hurwitz-Hecke 不变子簇

Hurwitz-Hecke Invariant Subvarieties

本文提出一种基于纯组合输入和 Hurwitz 空间理论在平移曲面分层中构造仿射不变子簇的方法，并证明其可用于构造 Bouw-Moeller Teichmueller 曲线及七个 Eskin-McMullen-Mukamel-Wright 秩二轨道闭包。

曲线模空间的全测地子簇与线性系统

Totally geodesic subvarieties of the moduli space of curves and linear systems

本研究在曲线模空间全测地子簇的一般曲线上构造线性系统，推导了维数≥2的全测地子簇的秩界，并完成了至多含两个零点的层中此类子簇的分类。

Reconstructing orbit closures from their boundaries

We introduce and study diamonds of GL(2,R)-invariant subvarieties of Abelian and quadratic differentials, which allow us to recover information on an invariant subvariety by simultaneously considering two degenerations, and which provide a new tool for the classification of invariant subvarieties. We classify a surprisingly rich collection of diamonds where the two degenerations are contained in trivial invariant subvarieties. Our main results have been applied to classify large collections of invariant subvarieties; the statement of those results do not involve diamonds, but their proofs rely…

Invariant subvarieties with small dynamical degree

Let $f:X\to X $ be a dominant self-morphism of an algebraic variety over an algebraically closed field of characteristic zero. We consider the set $Σ_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in $Σ_{f^{\infty}}$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\infty}}$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with small dynamical degree, and give an optimal upper bound (of cardinality) $$\sharp P_{f^n}\le d_1(f)^n(1+o(1))$$ as $n\to…

Generalizations of the Eierlegende-Wollmilchsau

We classify a natural collection of GL(2,R)-invariant subvarieties, which includes loci of double covers, the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces, and loci appearing naturally in the study of the complex geometry of Teichmuller space. This classification is the key input in subsequent work of the authors that classifies "high rank" invariant subvarieties, and in subsequent work of the first author that classifies certain invariant subvarieties with "Lyapunov spectrum as degenerate as possible". We also derive applications to the complex geometry o…

Algebraicity criteria, invariant subvarities and transcendence problems from arithmetic dynamics

We introduce an algebraicity criteria. It has the following form: under certain conditions, an analytic subvariety of some algebriac variety over a global field $K$, if it contains many $K$-points, then it is algebraic over $K.$ This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates. Combing this criteria and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamic…

GL(2,R)-Invariant Measures in Marked Strata: Generic Marked Points, Earle-Kra for Strata, and Illumination

We classify GL(2,R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmuller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2,R) orbit.

Invariant measures on finite rank subshifts

In this note we show that for any subshift $X$ of finite $S$-rank every invariant measure $μ$ is determined by its values on finitely many cylinders. Under mild conditions these cylinders are given by the letters of the alphabet in question.

Isoperiodic dynamics in rank 1 affine invariant orbifolds

Let M be a rank 1 affine invariant orbifold in a stratum of the moduli space of flat surfaces. We show that the leaves of the M-isoperiodic foliation are either all closed or all dense. In the second case, we establish ergodicity of the foliation with respect to the affine measure on M.

Invariant escaping Fatou components with two rank 1 limit functions for automorphisms of $\mathbb{C}^2$

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 1. 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.