Tien-Cuong Dinh, Nessim Sibony
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.
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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.
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 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 $E$.
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。
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.
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。
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.
$\log$-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。
We consider a meromorphic family of endomorphisms of degree at least 2 of a complex projective space that is parameterized by the unit disk.We prove that the measure of maximal entropy of these endomorphisms converges to the equilibrium measure of the associated non-Archimedean dynamical system when the system degenerates. The convergence holdsin the hybrid space constructed by Berkovich and further studied by Boucksom and Jonsson. We also infer from our analysis an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of endomorphisms.
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 …
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 characterize possible periodic subvarieties for surjective endomorphisms of complex abelian varieties in terms of the eigenvalues of the cohomological actions induced by the endomorphisms, extending previous work in this direction by Pink and Roessler. By applying our characterization to induced endomorphisms on Albanese varieties, we draw conclusions about the dynamics of surjective endomorphisms for a broad class of projective varieties. We also analyze several classes of surjective endomorphisms that are distinguished by properties of their cohomological actions.
We introduce the notion of an adelically summable normalized weight $g$, which is a family of normalized weights on the Berkovich projective lines satisfying a summability condition. We then establish an adelic equidistribution of effective $k$-divisors on the projective line over the separable closure $k_s$ in $\overline{k}$ of a product formula field $k$ having small $g$-heights and small diagonals. This equidistribution result generalizes Ye's for the Galois conjugacy classes of algebraic numbers with respect to quasi-adelic measures.
We introduce a new class of adelic heights on the projective line. We estimate their essential minimum and prove a result of equidistribution (at every place) for points of small height with estimates on the speed of convergence. To each rational function R in one variable and defined over a number field K, is associated a normalized height on the algebraic closure of K. We show that these dynamically defined heights are adelic in our sense, and deduce from this equidistribution results for preimages of points under R at every place of K. Our approach follows that of Bilu, and relies on potent…
In the present article, we define a notion of good height functions on quasi-projective varieties $V$ defined over number fields and prove an equidistribution theorem of small points for such height functions. Those good height functions are defined as limits of height functions associated with semi-positive adelic metrization on big and nef $\mathbb{Q}$-line bundles on projective models of $V$ satisfying mild assumptions. Building on a recent work of the author and Vigny as well as on a classical estimate of Call and Silverman, and inspiring from recent works of Kühne and Yuan and Zhang, we d…
Dynamically improper hypersurfaces for endomorphisms of projective space
本文推广预周期超曲面概念，并证明射影空间非线性自同态中具有周期临界点的映射在Zariski意义下稠密，解答了Ingram的疑问。
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…
A polarizable endomorphism on a projective variety enables us to consider given morphism as constant multiplication in the height function. In this paper, we will generalize it for arbitrary dominant endomorphism by defining the height expansion and contraction coefficients.
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…
Etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearly etale rational endomorphisms of weak Calabi-Yau varieties.
We say (W, \{ϕ_1,..., ϕ_t\}) is a polarizable dynamical system of several morphisms if ϕ_i are endomorphisms on a projective variety $W$ such that \bigotimes ϕ_i^*L is linearly equivalent to L^q} for some ample line bundle L on W and for some q>t. If q is a rational number, then we have the equidistribution of small points of given dynamical system because of Yuan's work. As its application, we can build a polarizable dynamical system of an automorphism and its inverse on $K3$ surface and show its periodic points are equidistributed.
Birational conjugacies between endomorphisms on the projective plane
本文证明了特征0下射影平面上次数至少为2的支配自同态若双有理共轭则必自同构共轭，并给出正特征反例。
We consider surjective endomorphisms f of degree > 1 on projective manifolds X of Picard number one and their f^{-1}-stable hypersurfaces V, and show that V is rationally chain connected. Also given is an optimal upper bound for the number of f^{-1}-stable prime divisors on (not necessarily smooth) projective varieties.
Let $Λ$ be a quasi-projective variety and assume that, either $Λ$ is a subvariety of the moduli space $\mathcal{M}_d$ of degree $d$ rational maps, or $Λ$ parametrizes an algebraic family $(f_λ)_{λ\inΛ}$ of degree $d$ rational maps on $\mathbb{P}^1$. We prove the equidistribution of parameters having $p$ distinct neutral cycles towards the $p$-th bifurcation current letting the periods of the cycles go to $\infty$, with an exponential speed of convergence. We deduce several fundamental consequences of this result on equidistribution and counting of hyperbolic components. A key step of the proof…
It is expected that a totally invariant divisor of a non-isomorphic endomorphism of the complex projective space is a union of hyperplanes. In this paper, we compute an upper bound for the degree of such a divisor. As a consequence, we prove the linearity of totally invariant divisors with isolated singularities.
Let $X$ be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $SEnd(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth, we show: there is a finite-index submonoid $G$ of $SEnd(X)$ such that $G$ acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group $Aut(X)$…
Lower Bounds for Galois Orbits of periodic points for polarized endomorphisms
本文研究极化内态射下n周期点的Galois轨道下界，并证明其等分布率的定量速率。
Let $f$ be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or stabilizer group, of a given $f$ for this action is known to be a finite group. We determine explicit families that parameterize all endomorphisms defined over $\bar{\mathbb{Q}}$ of degree $3$ and degree $4$ that have a nontrivial automorphism, the \textit{automorphism locus} of the moduli space of dynamical systems. We analyze the geometry of these loci in the appropriate moduli space of dynamical sys…
On smooth rationally connected projective threefolds of Picard number two admitting int-amplified endomorphisms
该研究证明Picard数为2的光滑有理连通射影三维簇为环面簇的充要条件是存在int-amplified自同态，并得出爆破等变下$\u202e\bb{P}^3$非同构满自同态的全不变光滑曲线必为直线的推论。
Let $X$ be a normal projective variety. A surjective endomorphism $f:X\to X$ is int-amplified if $f^\ast L - L =H$ for some ample Cartier divisors $L$ and $H$. This is a generalization of the so-called polarized endomorphism which requires that $f^*H\sim qH$ for some ample Cartier divisor $H$ and $q>1$. We show that this generalization keeps all nice properties of the polarized case in terms of the singularity, canonical divisor, and equivariant minimal model program.
We determine when the equidistribution property for possibly moving targets holds for a rational function of degree more than one on the projective line over an algebraically closed field of any characteristic and complete with respect to a non-trivial absolute value. This characterization could be useful in the positive characteristic case. Based on the variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to Favre and Rivera-Letelier, using a dynamical Diophantine approximation theorem by Silverman and by Szpiro--T…
We give a criterion for the weak convergence of unit Borel measures on the N-dimensional Berkovich projective space over a complete non-archimedean field. As an application, we give a sufficient condition for equidistribution in terms of a strong Zariski-density property on the scheme-theoretic projective space over the residue field. As a second application, in the case of residue characteristic zero we give an ergodic-theoretic equidistribution result for the powers of a point in the N-dimensional unit torus. This is a non-archimedean analogue of a well-known complex equidistribution result …
We study the dynamics of polynomial mappings f:C^k to C^k of degree at least 2 that extend continuously to projective space P^k. Our approach is to study the dynamics near the hyperplane at infinity and then making a descent to K --- the set of points with bounded orbits --- via external rays.
Let $X$ be a normal projective variety admitting a polarized or int-amplified endomorphism $f$. We list up characteristic properties of such an endomorphism and classify such a variety from the aspects of its singularity, anti-canonical divisor and Kodaira dimension. Then we run the equivariant minimal model program with respect to not just the single $f$ but also the monoid $SEnd(X)$ of all surjective endomorphisms of $X$, up to finite-index. Several applications are given. We also give both algebraic and geometric characterizations of toric varieties via polarized endomorphisms.
We consider surjective endomorphisms f of degree > 1 on the projective n-space with n = 3, and f^{-1}-stable hypersurfaces V. We show that V is a hyperplane (i.e., deg(V) = 1) but with four possible exceptions; it is conjectured that deg(V) = 1 for any n > 1.
We establish a quantitative adelic equidistribution theorem for a sequence of algebraic zeros divisors on the projective line over the separable closure of a product formula field having small diagonals and small $g$-heights with respect to an adelic normalized weight $g$ in arbitrary characteristic and in possibly non-separable setting, and obtain local proximity estimates between the iterations of a rational function $f\in k(z)$ of degree $>1$ and a rational function $a\in k(z)$ of degree $>0$ over a product formula field $k$ of characteristic $0$, applying this quantitative adelic equidistr…
We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mo…
On endomorphisms of projective varieties with numerically trivial canonical divisors
本文研究具有数值平凡典范除子的klt射影簇的拟放大与放大自同态，证明经迭代和等变双有理收缩拟放大自同态可下降为放大自同态，给出超凯勒簇、阿贝尔簇上相关自同态的判别与周期点性质。
We consider the potential density of rational points on an algebraic variety defined over a number field $K$, i.e., the property that the set of rational points of $X$ becomes Zariski dense after a finite field extension of $K$. For a non-uniruled projective variety with an int-amplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an int-amplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we p…
The holomorphic endomorphism f of projective space is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of f, has algebraic support (i.e., is a finite union of hypersurfaces). In the case of dimension 1, a deep result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattes maps. This note proves a corresponding result in arbitrary dimension, for a certain subclass of morphism. Specifically, we restrict attention to morphisms f of degree at least two, with…
A uniform Tits alternative for endomorphisms of the projective line
本文推广了投影直线自同态半群的Tits二择性，通过扩展算术高度函数的乒乓球论证，证明了指数增长半群具有一致指数增长。
Let $f \colon X \to X$ be a surjective endomorphism of a normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure of $X$. Using this, we extend the second author's result to singular surfaces to the extent that either $X$ has an $f$-invariant non-constant rational function, or $f$ has infinitely many Zariski-dense forward orbits; this result is also extended to Adelic topology (which is finer than Zariski topology).
We show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT($-1$) or rank-one CAT($0$) spaces, also hold for rank-one properly convex projective structures, equipped with their Hilbert metrics, admitting finite Sullivan measures built from appropriate conformal densities. In particular, this includes geometrically finite convex projective structures. More specifically, with respect to the Sullivan measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptot…
In this paper we prove the following theorem. Let $f$ be a dominant endomorphism of a smooth projective surface over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then there exists a closed point whose orbit under $f$ is Zariski dense. This result gives us a positive answer to the Zariski dense orbit conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang, for endomorphisms of smooth projective surfaces. Moreover, we define a new canonical topology on varieties over an algebraically cl…
On the centralizers of endomorphisms of the projective line
本文研究了射影线端射影的中心化子，并证明其并集等于某个迭代的中心化子，解决了Pakovich的猜想并扩展了相关半群的蒂斯替代定理。
Fix a finite field. A hyperelliptic curve determines a measure on the discrete space of rank two bundles on the projective line: the mass of a given vector bundle is the number of line bundles whose pushforward it is. In a sequence of hyperelliptic curves whose genera tend to infinity, these measures tend to the natural measure on the space of rank two bundles. This is a function field analogue of Duke's theorem on the equidistribution of Heegner points, and can be proven similarly: it follows from a manipulation of zeta functions, plus the Riemann Hypothesis for curves. Likewise, a sequence o…
For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, we prove that there exists a Zariski dense set of rational points h…
It is an open problem whether repelling periodic points are dense in the classical Julia set of a non-archimedean rational function of degree more than one. We give a partial positive answer to this question based on a study of a logarithmic equidistribution on the Berkovich projective line over non-archimedean fields.
It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described by methods of equivariant lifting and fibrations. (For endomorphisms of projective spaces, see version 1).
Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms
针对Broustet和Gongyo提出的容许极化自同态的正规射影簇为Calabi-Yau型的猜想，研究者基于等变极小模型纲领与典范丛公式建立框架，证明了光滑射影三维簇情形下该猜想成立。
Kawaguchi-Silverman conjecture for certain surjective endomorphisms
本文证明Kawaguchi-Silverman猜想对任意射影曲面及若干高维簇（如有理连通光滑三维簇）的满射自同态成立，核心方法结合等变极小模型纲领、反典范除子有效性与环面偶对刻画。
There is a natural conjugation action on the set of endomorphism of $¶^N$ of fixed degree $d \geq 2$. The quotient by this action forms the moduli of degree $d$ endomorphisms of $¶^N$, denoted $\mathcal{M}_d^N$. We construct invariant functions on this moduli space coming from to set of multiplier matrices of the periodic points. The basic properties of these functions are demonstrated such as that they are in the ring of regular functions of $\mathcal{M}_d^N$, methods of computing them, as well as the existence of relations. The main part of the article examines to what extend these invariant…
Structures theorems and applications of non-isomorphic surjective endomorphisms of smooth projective threefolds
本文研究了光滑射影三维簇的非同构满射自同态，证明了在$δ$-本原条件下双有理极小模型纲领迭代后变为$f$-等变，并据此证明了Kawaguchi-Silverman猜想及将Zariski稠密轨道猜想归结为终端三维簇情形。
An endomorphism $f$ of a projective variety X is polarized (resp. quasi-polarized) if $f^*H$ is linearly equivalent to $qH$ for some ample (resp. nef and big) Cartier divisor $H$ and integer $q > 1$. First, we use cone analysis to show that a quasi-polarized endomorphism is always polarized, and the polarized property descends via any equivariant dominant rational map. Next, we show that a suitable maximal rationally connected fibration (MRC) can be made $f$-equivariant using a construction of N. Nakayama, that $f$ descends to a polarized endomorphism of the base Y of this MRC and that this Y …
Let g be a nonconstant rational map from the projective line to itself that has degree greater than one and is defined over a number field. The map g gives rise to generalized Mahler measures for polynomials in one variable. We use diophantine approximation to show that the generalized Mahler measure of a polynomial F at a place v can be computed by averaging the log of the v-adic absolute value of F over the periodic points of g. This allows us to compute canonical heights of algebraic points via equidistribution on periodic points.
Let $K$ be a number field and $v$ a non archimedean valuation on $K$. We say that an endomorphism $Φ\colon \mathbb{P}_1\to \mathbb{P}_1$ has good reduction at $v$ if there exists a model $Ψ$ for $Φ$ such that $\degΨ_v$, the degree of the reduction of $Ψ$ modulo $v$, equals $\degΨ$ and $Ψ_v$ is separable. We prove a criterion for good reduction that is the natural generalization of a result due to Zannier in \cite{Uz3}. Our result is in connection with other two notions of good reduction, the simple and the critically good reduction. The last part of our article is dedicated to prove a characte…
We study the dynamics of stochastic families of rational maps on the projective line. As such families can be infinite and may not typically be defined over a single number field, we introduce the concept of generalized adelic measures, generalizing previous notions introduced by Favre and Rivera-Letelier and Mavraki and Ye. Generalized adelic measures are defined over the measure space of places of an algebraic closure of the rationals, using a framework established by Allcock and Vaaler. This turns our heights from sums over places into integrals. We prove an equidistribution result for gene…
Non-projective K3 surfaces with real or Salem multiplication
本文研究非投影复K3曲面的Hodge内射代数，证明其为完全实数域或由萨姆尔数生成的数域，并探讨其在数论和动力学中的应用。
Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures
本文研究了拟射影簇上有限满射自同态的Zariski稠密轨道猜想（ZDO）、Adelic Zariski稠密轨道猜想（AZO）及Kawaguchi-Silverman猜想（KSC），并在光滑仿射曲面、射影簇和半阿贝尔簇等特定条件下证明了这些猜想成立。
In this work, we study dominant rational maps preserving singular holomorphic codimension one foliations on projective manifolds and that exhibit non-trivial transverse dynamics.
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