Tien-Cuong Dinh, Nessim Sibony
Let f be a non-invertible holomorphic endomorphism of a projective space and f^n its iterate of order n. We prove that the pull-back by f^n of a generic (in the Zariski sense) hypersurface, properly normalized, converge to the Green current associated to f when n tends to infinity. We also give an analogous result for the pull-back of positive closed (1,1)-currents.
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Equidistribution of graphs of holomorphic correspondences
研究紧黎曼曲面上的全纯自对应，在动力学度不相等或非弱模条件下，证明迭代图的指数快速等分布。
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 $E$.
Let f be a non-invertible holomorphic endomorphism of P^k. For a hypersurface H of P^k, generic in the Zariski sense, we give an explicit speed of convergence of f^{-n}(H) towards the dynamical Green (1,1)-current of f.
Equidistribution of saddle periodic points for automorphisms on compact Kähler manifolds
本文证明了紧致Kähler流形上具有简单上同调作用的全纯自同构的鞍型周期点关于平衡测度等分布，并推广到正拓扑熵的Kähler曲面情形。
$\log$-Hölder regularity of currents and equidistribution towards Green currents
本文研究了在 $\log$-Hölder-连续可观测条件下，动力系统迭代拉回当前对 Green 当前的指数收敛性。
Equidistribution towards the Green current in big cohomology classes
本文研究了紧致凯勒流形上亚纯自映射的光滑(1,1)形式和正闭电流的拉回序列的极限分布，探讨了其在大上同调类中的等分布性质。
We establish an approximation of the activity current $T_c$ in the parameter space of a holomorphic family $f$ of rational functions having a marked critical point $c$ by parameters for which $c$ is periodic under $f$, i.e., is a superattracting periodic point. This partly generalizes a Dujardin--Favre theorem for rational functions having preperiodic points, and refines a Bassanelli--Berteloot theorem on a similar approximation of the bifurcation current $T_f$ of the holomorphic family $f$. The proof is based on a dynamical counterpart of this approximation.
Given a holomorphic self-map of complex projective space of degree larger than one, we prove that there exists a finite collection of totally invariant algebraic sets with the following property: given any positive closed (1,1)-current of mass 1 with no mass on any element of this family, the sequence of normalized pull-backs of the current converges to the Green current. Under suitable geometric conditions on the collection of totally invariant algebraic sets, we prove a sharper equidistribution result.
In this article, we consider currents given by the p-fold non-pluripolar product associated with a quasi-plurisubharmonic function of finite energy, and prove that normalized pull-backs of such currents converge to the Green (p, p)-current exponentially fast in the sense of currents for holomorphic endomorphisms of Pk.
Let f be a non-invertible holomorphic endomorphism of the complex projective space P^k and f^n its iterate of order n. Let V be an algebraic subvariety of P^k which is generic in the Zariski sense. We give here a survey on the asymptotic equidistribution of the sequence $f^{-n}(V)$ when n goes to infinity.
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 $\mathbb{C}^k$.
In this paper we review the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P^1 or subgroups of the Moebius group PSL(2,C)). The topics covered include: the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties on parameter space.
Let f be a non-invertible holomorphic endomorphism of the complex projective space P^k, f^n its iterate of order n and μthe equilibrium measure of f. We estimate the speed of convergence in the following known result. If a is a Zariski generic point in P^k, the probability measures, equidistributed on the preimages of a under f^n, converge to μas n goes to infinity.
Equidistribution of iterations of holomorphic correspondences and Hutchinson invariant sets
本文研究全纯对应的迭代等分布性质，证明关联多值映射的迭代像收敛到满足特定不变关系的概率测度，并据此证明一类算子在充分大次数下存在极小Hutchinson不变集，且在附加条件下该集合为Cantor集。
Equidistribution for non-pluripolar currents on compact Kähler manifolds
本文研究了紧凯勒流形上具有简单上同调作用的满射全纯自同态，证明了非多重极电流在迭代拉回下的归一化序列收敛到格林电流，揭示了动力系统与多重位势理论之间的深层联系。
We initiate a parametric study of holomorphic families of polynomial skew products, i.e., polynomial endomorphisms of $\mathbb{C}^2$ of the form $F(z,w)= (p(z), q(z,w))$ that extend to holomorphic endomorphisms of $\mathbb{P}^2(\mathbb{C})$. We prove that dynamical stability in the sense of arXiv:1403.7603 preserves hyperbolicity within such families, and give a complete classification of the hyperbolic components that are the analogue, in this setting, of the complement of the Mandelbrot set for the family $z^2 +c$. We also precisely describe the geometry of the bifurcation locus and current …
We extend Holowinsky and Soundararajan's proof of quantum unique ergodicity for holomorphic Hecke modular forms on SL(2,Z), by establishing it for automorphic forms of cohomological type on GL_2 over an arbitrary number field which satisfy the Ramanujan bounds. In particular, we have uncondtional theorems over totally real and imaginary quadratic fields. In the totally real case we show that our result implies the equidistribution of the zero divisors of holomorphic Hecke modular forms, generalising a result of Rudnick over Q.
Equidistribution speed of iterated preimages for rational maps on the Riemann sphere
本文研究了有理映射迭代预像的正则分布速度，针对非超吸引周期点在一维情况下获得了接近最优的$O(nd^{-n})$速度，并推广至Hölder连续观测函数，同时对几何有限有理映射证明了更优的$O(d^{-n})$速度。
We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to t…
Equidistribution of saddle periodic points for Hénon-like maps
本文证明在动力度自然假设下，任意维Hénon型映射的鞍周期点关于平衡测度等分布，推广了前人的结果并改进其技术，同时研究Green流的精细性质。
Green currents of holomorphic correspondences on compact Kähler manifolds
研究全纯对应物在紧致凯勒流形上的绿色流构造及其超势的连续性，证明正闭流的指数均衡分布特性。
Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$
本文证明了 $\mathbb P^k$ 的全纯自同态的周期点指数级快速等分布于平衡测度，量化了 Lyubich 和 Briend-Duval 的定理，并揭示了小 Julia 集中存在大量具有大乘子的周期循环。
Equidistribution without stability for toric surface maps
本文针对无法双有理共轭到代数稳定映射的一类环面曲面有理映射，证明了曲线迭代原像的等分布结果，并在小拓扑度条件下证明了迭代前向像的等分布，为此系统发展了环面曲面逆向极限上的正闭(1,1)流理论框架。
We study the laminarity of the Green current of endomorphisms of $P^2C$ near hyperbolic measures of saddle type. When these measures are supported by attracting sets, we prove that the Green current is laminar in the basin of attraction and we obtain new ergodic properties. This generalizes some results of Bedford and Jonsson on regular polynomial mappings in $C^2$.
Quantitative equidistribution of periodic points for rational maps
本文证明了复有理映射的周期点以显式速率定量均匀分布于平衡测度，并借助 Green 函数正则性、Hsia 核估计及 Moriwaki 推广的乘积公式完成证明。
Let $K$ be a complete, algebraically closed, non-Archimedean valued field, and let $ϕ\in K(z)$ with $\textrm{deg}(ϕ) \geq 2$. In this paper we consider the family of functions $\textrm{ordRes}_{ϕ^n}(x)$, which measure the resultant of $ϕ^n$ at points $x$ in $\textbf{P}^1_{\textrm{K}}$, the Berkovich projective line, and show that they converge locally uniformly to the diagonal values of the Arakelov-Green's function $g_{μ_ϕ}(x,x)$ attached to the canonical measure of $ϕ$. Following this, we are able to prove an equidistribution result for Rumely's crucial measures $ν_{ϕ^n}$, each of which is a…
Well-distribution of Polynomial maps on locally compact groups
本文推广了Weyl及Green和Tao等人的经典一致分布定理，提出并证明了从局部紧群到紧阿贝尔群的多项式映射的一般一致分布定理。
Mass equidistribution for lifts on hyperbolic $4$-manifolds
本文证明了超几何4流形上Pitale提升序列的量子唯一混浊性猜想，通过构造具有有利几何性质的放大器，首次成功应用于逃离非 tempered 子群的情形。
Let $X$ be a compact Kähler manifold of dimension 3 and let $f:X\rightarrow X$ be a pseudo-automorphism. Under the mild condition that $λ_1(f)^2>λ_2(f)$, we prove the existence of invariant positive closed $(1,1)$ and $(2,2)$ currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equidistribution result (which is essentially known in the literature) for Green $(1,1)$ currents of meromorphic selfmaps, not necessarily 1-algebraic stable, of a compact Kähler manifold of arbitrary dimension; and discuss how a stronger equidistribution result may be…
We study random holomorphic endomorphisms of P^k(C). Under some assumptions, we construct a random Green current and a random Green measure and we prove that these measures have mixing properties.
We establish a formula for the sum of the Lyapounov exponents of an holomorphic endomorphism of ${\bf P}^k$. For an holomorphic family of such endomorphisms we define the {\em bifurcation current} as $dd^cL$ and show that it vanishes when the repulsive cycles move holomorphically. We then prove a formula which relates this current with the interaction between the Green current and the current of integration on the critical set. In the 1-dimensional case (i.e. for ${\bf P}^1$) we find a geometrical description of the support of this current and its powers. Finally we introduce the {\em bifurcat…
Equidistribution for matings of quadratic maps with the Modular group
研究二次有理映射与模群配对的全纯对应族的渐近行为，证明非例外点的迭代像与原像及周期点均等分布，即使该族在特定意义下仅为弱模而非模。
We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.
A holomorphic endomorphism f of CP^2 admits a Julia set J_1, defined as usual to be the locus of non-normality of its iterates, and a (typically) smaller Julia set J_2, which is essentially the closure of the set of repelling periodic orbits. The question has been raised whether J_1\J_2 is filled (possibly in a measure-theoretic sense) with "Fatou subvarieties" along which the dynamics is locally equicontinuous. In this article we construct examples showing that this is not the case in general
Let f an holomorphic endomorphism of CP(k) with degree larger than 2. We show that if the Green measure of f is not singular, then f is rigid : it is a Lattes example. The proof relies on a linearization property of the iterates of f, along typical orbits. This property allow us to regularize the Green current, and to prove the rigidity. -- -- -- Soit f un endomorphisme holomorphe de CP(k) de degre plus grand que 2. Nous montrons que si la mesure de Green de f n'est pas singuliere, alors f est tres rigide : c'est un exemple de Lattes. La demonstration repose sur une propriete de linearisation …
In the first part of the thesis, we study some dynamical properties of skew products of Hénon maps of $\mbb C^2$ that are fibered over a compact metric space $M$. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of Hénon mappings. In analogy with the dynamics of the iterates of a single Hénon map, it is possible to construct fibered Green functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. Further, it is shown that the successive pullbacks of a suitable current by the skew Hénon maps…
We prove that the Julia set of a Henon type automorphism on C^2 is very rigid: it supports a unique positive ddc-closed current of mass 1. A similar property holds for the cohomology class of the Green current associated with an automorphism of positive entropy on a compact Kaehler surface. Relations between this phenomenon, several quantitative equidistribution properties and the theory of value distribution will be discussed. We also survey some rigidity properties of Henon type maps on C^k and of automorphisms of compact Kaehler manifolds.
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。
We show that the main part of the genus of the preimages of a projective line by a generic holomorphic endomorphism of CP^2 goes to the support of the Green measure.
In this paper, we set up a "dictionary" between discrete Schrödinger operators and the holomorphic dynamics on certain affine cubic surfaces, building on previous work by Cantat, Damanik and Gorodetski. Paragraphs 3.2.2 and 3.2.3 were written by Serge Cantat as an appendix.
We show that the Lattes endomorphisms are the only holomorphic endomorphisms of the complex k-dimensional projective space whose measure of maximal entropy is absolutely continuous with respect to the Lebesgue measure. As a consequence, Lattes endomorphisms are also characterized by other extremal properties as the maximality of the Hausdorff dimension of their measure of maximal entropy or the minimality of their Liapounov exponents. Our proof uses a linearization method which is of independant interest and a previous characterization by the regularity of the Green current.
The emphasis of this course is on pluripotential methods in complex dynamics in higher dimension. They are based on the compactness properties of plurisubharmonic functions and on the theory of positive closed currents. Applications of these methods are not limited to the dynamical systems that we consider here. We choose to show their effectiveness and to describe the theory for two large families of maps. The first chapter deals with holomorphic endomorphisms of the projective space P^k. We establish the first properties and give several constructions for the Green currents and the equilibri…
Let f be a holomorphic automorphism of a compact Kahler manifold (X,ω) of dimension k>1. We study the convex cones of positive closed (p,p)-currents T_p, which satisfy a functional relation $f^*(T_p)=λT_p, λ>1,$ and some regularity condition (PB). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropy h(f) of f is positive, then for some m, there is a positive closed (m,m)-current T_m which satisfies the relation $f^* T_m=\exp(h(f)) T_m$. Moreover,every quasi-p.s.h. func…
Let $Λ$ be a complex manifold and let $(f_λ)_{λ\in Λ}$ be a holomorphic family of rational maps of degree $d\geq 2$ of $\mathbb{P}^1$. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdin's bound of the volume of the image of a dynamical ball. Applying our technics to complex dynamics …
We study some dynamical properties of skew products of Hénon maps of $\mbb C^2$ that are fibered over a compact metric space $M$. The problem reduces to understanding the dynamical behavior of the composition of a pseudo-random sequence of Hénon mappings. In analogy with the dynamics of the iterates of a single Hénon map, it is possible to construct fibered Green's functions that satisfy suitable invariance properties and the corresponding stable and unstable currents. This analogy is carried forth in two ways: it is shown that the successive pullbacks of a suitable current by the skew Hénon m…
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