Structure properties of laminar currents on $\mathbb{P}^2$

Wermer examples and currents

In this paper we give the first examples of positive closed currents in $\mathbb{C}^2$ with continuous potentials, vanishing self-intersection, and which are not laminar. More precisely, they are supported on sets "without analytic structure". The result is mostly interesting when the potential has regularity close to $C^2$, because laminarity is expected to hold in that case. We actually construct examples which are $C^{1,α}$ for all $α<1$.

Approximation des fonctions lisses sur certaines laminations

We show that on a Riemann surface lamination locally embedded in $\mathbb{C}^2$, $C^1$ functions (in the sense of the $C^1$ structure of the lamination) are uniform limits of ambient $C^1$ functions, with $L^p$ control on the derivatives along the leaves. This implies that locally in $C^2$, a (1,1) positive closed current dominated by a uniformly laminar current is itself uniformly laminar.

Parametric Lyapunov exponents

In an algebraic family of rational maps of $\mathbb{P}^1$, we show that, for almost every parameter for the trace of the bifurcation current of a marked critical value, the critical value is Collet-Eckmann. This extends previous results of Graczyk and Świcatek in the unicritical family, using Makarov theorem. Our methods are based instead on ideas of laminar currents theory.

Structure of Julia sets for post-critically finite endomorphisms on $\mathbb{p}^2$

Let $f$ be a post-critically finite endomorphism (PCF map for short) on $\mathbb{P}^2$, let $J_1$ denote the Julia set and let $J_2$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $J_1\setminus J_2$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $x\in J_2$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing $x$. Here sporadic means that the super-saddle cycle is not contained in a critical co…

Directional dimensions of ergodic currents on $\mathbb C \mathbb P (2)$

LLet $f$ be a holomorphic endomorphism of $\mathbb P^ 2$ of degree $d \geq 2$. We estimate the local directional dimensions of closed positive currents $S$ with respect to ergodic dilating measures $ν$. We infer several applications. The first one shows that the currents $S$ containing a measure of entropy $h\_ν> \log d$ have a directional dimension $>2$, which answers a question by de Thélin-Vigny. The second application asserts that the Dujardin's semi-extremal endomorphisms are close to suspensions of one-dimensional Lattès maps. Finally, we obtain an upper bound for the dimension of the eq…

Local regularity of super-potentials and equidistribution of positive closed currents of $\mathbb{P}^k$

In this paper, we introduce notions about local regularity of the super-potential and prove equidistribution of positive closed $(p, p)$-currents of $\mathbb{P}^k$ whose super-potentials are bounded near an invariant analytic subset, for holomorphic endomorphisms of $\mathbb{P}^k$ and for regular polynomial automorphisms of $\mathbb{C}^k$.

Directed harmonic currents for laminations on certain compact complex surfaces

Let $\mathcal{L}$ be a Lipschitz lamination by Riemann surfaces embedded in $M$. If $M$ is a complex torus, $\mathbb{CP}^1\times\mathbb{CP}^1$ or $\mathbb{T}^1\times\mathbb{CP}^1$ and there is no directed closed current then there exists a unique directed harmonic current of mass one. Moreover if $\mathcal{L}$ is embedded in $M=\mathbb{CP}^1\times\mathbb{CP}^1$ and has no compact leaves, then there is no directed closed current. If $\mathcal{L}$ is not Lipschitz, then slightly weaker results are obtained.

Attracting Currents and Equilibrium Measures for Quasi-attractors of $\mathbb P^k$

Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of degree $d.$ For each quasi-attractor of $f$ we construct a finite set of currents with attractive behaviors. To every such an attracting current is associated an equilibrium measure which allows for a systematic ergodic theoretical approach in the study of quasi-attractors of $\mathbb P^k.$ As a consequence, we deduce that there exist at most countably many quasi-attractors, each one with topological entropy equal to a multiple of $\log d.$ We also show that the study of these analytic objects can initiate a bifurcation theory for attra…

Equidistribution in Higher Codimension for Holomorphic Endomorphisms of $\mathbb{P}^k$

In this paper, we discuss the equidistribution phenomena for holomorphic endomorphisms over $\mathbb{P}^k$ in the case of bidegree $(p,p)$ with $1<p<k$. We prove that if $f:\mathbb{P}^k\to\mathbb{P}^k$ is a holomorphic endomorphism of degree $d\geq 2$ and $T^p$ denotes the Green $(p,p)$-current associated with $f$, then there exists a proper invariant analytic subset $E$ for $f$ such that $d^{-pn}(f^n)^*(S)\to T^p$ exponentially fast in the current sense for every positive closed $(p,p)$-current $S$ of mass 1 such that $S$ is smooth on $E$.

Pluripotential theory on the support of closed positive currents and applications to dynamics in $\mathbb{C}^n$

We extend certain classical theorems in pluripotential theory to a class of functions defined on the support of a $(1,1)$-closed positive current $T$, analogous to plurisubharmonic functions, called $T$-plurisubharmonic functions. These functions are defined as limits, on the support of $T$, of sequences of plurisubharmonic functions decreasing on this support. In particular, we show that the poles of such functions are pluripolar sets. We also show that the maximum principle and the Hartogs's theorem remain valid in a weak sense. We study these functions by means of a class of measures, so-ca…

产生不同Nevanlinna流的整曲线

Entire curves producing distinct Nevanlinna currents

该研究受Sibony问题启发证明具特定Oka性质的紧复流形上存在生成所有Nevanlinna/Ahlfors流的整曲线，还正面解答Yau的问题，在两椭圆曲线乘积中构造出可产生无穷多不同Nevanlinna/Ahlfors流的整曲线，揭示整曲线的全纯灵活性。

On the dynamics near infinity of some polynomial mappings in $\mathbb{C}^2$

We construct the Green current for a random iteration of "horizontal-like" mappings in two complex dimensions. This is applied to the study of a polynomial map $f:\mathbb{C}^2\to\mathbb{C}^2$ with the following properties: 1. infinity is $f$-attracting, 2. $f$ contracts the line at infinity to a point not in the indeterminacy set. Then the Green current of $f$ can be decomposed into pieces associated with an itinerary determined by the indeterminacy points. We also study the set of escape rates near infinity, i.e. the possible values of the function $\limsup \frac{1}{n}\log^+\log^+ \norm{f^n}$…

On Ahlfors currents

We answer a basic question in Nevanlinna theory that Ahlfors currents associated to the same entire curve may be nonunique. Indeed, we will construct one exotic entire curve $f: \mathbb{C}\rightarrow X$ which produces infinitely many cohomologically different Ahlfors currents. Moreover, concerning Siu's decomposition, for an arbitrary $k\in \mathbb{Z}_{+}\cup \{\infty\}$, some of the obtained Ahlfors currents have singular parts supported on $k$ irreducible curves. In addition, they can have nonzero diffuse parts as well. Lastly, we provide new examples of diffuse Ahlfors currents on the produ…

Equilibrium states of endomorphisms of $\mathbb{P}^k$ I: existence and properties

We develop a new method, based on pluripotential theory, to study the transfer (Perron-Frobenius) operator induced on $\mathbb P^k = \mathbb P^k (\mathbb C)$ by a holomorphic endomorphism and a suitable continuous weight. This method allows us to prove the existence and uniqueness of the equilibrium state and conformal measure for very general weights (due to Denker-Przytycki-Urbański in dimension 1 and Urbański-Zdunik in higher dimensions, both in the case of H\''older continuous weights). We establish a number of properties of the equilibrium states, including mixing, K-mixing, mixing of all…

On Lelong Numbers of Positive Closed Currents on $\mathbb{P}^n$

Let $T$ be a positive closed current of bidimension $(p,p)$ with unit mass on the complex projective space $\mathbb P^n$. For certain values of $α$ and $β= β(p, α)$ we show that if $T$ has enough points where the Lelong number is at least $α$, then the upper level set $E_β^+ (T)$ of points where $T$ has Lelong number strictly larger than $β$ has certain geometric properties.

An equidistribution theorem for biraitonal maps of $\mathbb{P}^k$

We prove an equidistribution theorem of positive closed currents for a certain class of birational maps $f_+:\mathbb{P}^k\to\mathbb{P}^k$ of algebraic degree $d\geq 2$ satisfying $\bigcup_{n\geq 0}f_-^n(I^+)\cap \bigcup_{n\geq 0}f_+^n(I^-)=\emptyset$, where $f_-$ is the inverse of $f_+$ and $I^\pm$ are the sets of indeterminacy for $f_\pm$, respectively.

非双曲线性化奇点附近的有向调和流

Directed harmonic currents near non-hyperbolic linearized singularities

本文研究非双曲线性化奇点附近的有向调和流，证明了实参数λ不同情形下0点Lelong数的取值结论，补充了Nguyen关于双曲情形的已有结果。

A Property of Upper Level Sets of Lelong Numbers of Currents on $\mathbb{P}^2$

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on the complex projective space $\mathbb P^2$. For $α> 2/5$ and $β= (2-2α)/3$ we show that if $T$ has four point with Lelong number greater than $α$, the upper level set $E_β^+ (T)$ of points of $T$ with Lelong number strictly larger than $β$ is contained within a conic with the exception of at most one point.

生成Nevanlinna currents所有形状的整曲线

Entire curves generating all shapes of Nevanlinna currents

本文证明复环面中整曲线可生成所有Nevanlinna currents，并构造扭曲整曲线实现任意形状的Siu分解，填补缺失情形，回答Yau-Zhou问题，展现Oka几何中整曲线的全纯灵活性。

关于 CP(2) 上 Green 流的切片测度

On slice measures of Green currents on CP(2)

本文为 Dujardin 关于 CP(2) 上平衡测度绝对连续性与 Lyapunov 指数关系的定理提供新证明，并研究由全纯局部淹没定义的切片测度，给出其与平衡测度相等的充分条件。

Complex Tropical Currents, Extremality, and Approximations

To a tropical $p$-cycle $V_{\mathbb{T}}$ in $\mathbb{R}^n$, we naturally associate a normal closed and $(p,p)$-dimensional current on $(\mathbb{C}^*)^n$ denoted by $\mathscr{T}_n^p(V_{\mathbb{T}})$. Such a "tropical current" $\mathscr{T}_n^p(V_{\mathbb{T}})$ will not be an integration current along any analytic set, since its support has the form ${\rm Log\,}^{-1}(V_{\mathbb{T}})\subset (\mathbb{C}^*)^n$, where ${\rm Log\,}$ is the coordinate-wise valuation with $\log(|.|)$. We remark that tropical currents can be used to deduce an intersection theory for effective tropical cycles. Furthermore…

通过单条整曲线生成所有Ahlfors流

Generating all Ahlfors currents by a single entire curve

针对具有圆盘Runge逼近性质的紧复流形，构造单条整曲线，其关联的同心全纯圆盘族生成所有Ahlfors流，解决了Sibony猜想。

Polynomial interpolation and residue currents

We show that a global holomorphic section of $\mathscr{O}(d)$ restricted to a closed complex subspace $X \subset \mathbb{P}^n$ has an interpolant if and only if it satisfies a set of moment conditions that involves a residue current associated with a locally free resolution of $\mathscr{O}_X$. When $X$ is a finite set of points in $\mathbb{C}^n \subset \mathbb{P}^n$ this can be interpreted as a set of linear conditions that a function on $X$ has to satisfy in order to have a polynomial interpolant of degree at most $d$.

Résolution du $\partial\bar{\partial}$ pour les courants prolongeables définis sur un domaine pseudoconvexe non borné de $\mathbb{C}^n$

In this present paper, we solve the $\partial\bar{\partial}$ for extendable currents definite in a pseudoconvexe unbounded domain of $\mathbb{C}^n$ .

Lelong Numbers of Bidegree (1,1) Currents on Multiprojective Spaces

Let $T$ be a positive closed current of bidegree $(1,1)$ on a multiprojective space $X={\mathbb P}^{n_1}\times\ldots\times{\mathbb P}^{n_k}$. For certain values of $α$, which depend on the cohomology class of $T$, we show that the set of points of $X$ where the Lelong numbers of $T$ exceed $α$ have certain geometric properties. We also describe the currents $T$ that have the largest possible Lelong number in a given cohomology class, and the set of points where this number is assumed.

Super currents and tropical geometry

We introduce the formalism of positive super currents on \mathbb{R}^{n}, in strong analogy with the theory of positive currents in \mathbb{C}^{n}. We consider intersection of currents and Lelong numbers, and as an application we show that the formalism can be used to describe tropical varieties. This is similar in spirit to the fact that in complex analysis the current of integration of an analytic variety can be identified with a closed, positive current.

Minimal hulls of compact sets in $\mathbb R^3$

The main result of this paper is a characterization of the minimal surface hull of a compact set $K$ in $\mathbb R^3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense, and also by $2$-dimensional minimal currents which are limits of Green currents supported by conformal minimal discs. Analogous results are obtained for the null hull of a compact subset of $\mathbb C^3$. We also prove a null hull analogue of the Alexander-Stolzenberg-Wermer theorem on polynomial hulls of compact sets of finite linear measure, and a polynomial hull version o…

复数空间中具有指数分布的随机P多项式的零点

Zeros of random $P$-polynomials in $\mathbb{C}^d$ with exponential profiles

研究复数空间中具有指数分布的随机多变量P多项式的零点，证明归一化势函数收敛到确定性托里克 plurisubharmonic 函数，并讨论零点流的弱收敛性。

Upper level sets of Lelong numbers on $\mathbb P^2$ and cubic curves

Let $T$ be a positive closed current of bidimension $(1,1)$ with unit mass on $\mathbb P^2$ and $V_α(T)$ be the upper level sets of Lelong numbers $ν(T,x)$ of $T$. For any $α\geq \frac{1}{3}$, we show that $|V_α(T)\setminus C|\leq 2$ for some cubic curve $C$.

Lelong数的上水平集在Hirzebruch曲面上的研究

Upper level sets of Lelong numbers on Hirzebruch surfaces

本文研究Hirzebruch曲面上正闭电流的Lelong数上水平集的几何约束，证明其包含在特定低次数曲线中，并给出双次数与参数β的精确条件。

复化 horocycle 特征函数的增长与节点流

Growth and nodal current of complexified horocycle eigenfunctions

研究双曲平面上磁量子哈密顿量特征函数（horocycle 特征函数）的解析延拓在 Grauert 管中的增长与除子分布，通过微局部量子遍历性和虚时 horocycle 流建立渐近估计，揭示复化规范因子对增长的主导作用。