Dynamics of Automorphisms of Compact Complex Manifolds

On the automorphism group of rational manifolds

In this note, we prove that every automorphism of a rational manifold which is obtained from $\Bbb{P}^k$ by a finite sequence blow-ups along smooth centers of dimension at most r with k>2r+2 has zero topological entropy.

Compact Kähler manifolds with automorphism groups of maximal rank

For an automorphism group G on an n-dimensional (n > 2) normal projective variety or a compact Kähler manifold X so that G modulo its subgroup N(G) of null entropy elements is an abelian group of maximal rank n-1, we show that N(G) is virtually contained in Aut_0(X), the X is a quotient of a complex torus T and G is mostly descended from the symmetries on the torus T, provided that both X and the pair (X, G) are minimal.

Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups

Thanks to the theory of Coxeter groups, we produce the first family of Calabi-Yau manifolds $X$ of arbitrary dimension $n$, for which $\Bir(X)$ is infinite and the Kawamata-Morrison movable cone conjecture is satisfied. For this family, the movable cone is explicitly described; it's fractal nature is related to limit sets of Kleinian groups and to the Apollonian Gasket. Then, we produce explicit examples of (biregular) automorphisms with positive entropy on some Calabi-Yau manifolds.

Tits alternative for automorphism groups of compact Kaehler manifolds

We survey some properties of the automorphism groups of compact Kaehler manifolds. In particular, we present recent results by Keum, Oguiso and Zhang on the structure of these groups from the Tits alternative point of view. Several other related results will be also discussed. This paper is written for the VIASM Annual Meeting 2012.

低维紧复空间上的野自同构

Wild automorphisms of compact complex spaces of lower dimensions

本文分类了低维紧复空间上具有野自同构的对象，证明紧复曲面情形必为复环面或某类Inoue曲面且熵为零，并将Oguiso–Zhang关于射影情形的结果推广到紧Kähler三维与四维流形。

双Kähler流形族中双有理自同构群的代数熵

Algebraic entropy of birational automorphism groups in families of hyper-Kähler manifolds

本文研究射影双Kähler流形族中双有理自同构群的代数熵行为，探讨其性质与变化。

高维流形自同构群的约束

Constraints on automorphism groups of higher dimensional manifolds

本文研究了通过有限次吹胀得到的有理流形的自同构群，证明其在整数系数的上同调群中的像有限，并指出每个全纯自同构的拓扑熵为零。

A characterization of compact complex tori via automorphism groups

We show that a compact Kaehler manifold X is a complex torus if both the continuous part and discrete part of some automorphism group G of X are infinite groups, unless X is bimeromorphic to a non-trivial G-equivariant fibration. Some applications to dynamics are given.

超 Kähler 流形的抛物自同构

Parabolic automorphisms of hyperkahler manifolds

本文研究超 Kähler 流形上的抛物自同构，证明保持 Lagrangian 纤维化者在纤维上遍历作用，并将 K3 曲面的相关结果推广至一般超 Kähler 情形。

Compact Kähler manifolds admitting large solvable groups of automorphisms

Let G be a group of automorphisms of a compact Kähler manifold X of dimension n and N(G) the subset of null-entropy elements. Suppose G admits no non-abelian free subgroup. Improving the known Tits alternative, we obtain that, up to replace G by a finite-index subgroup, either G/N(G) is a free abelian group of rank < n-1, or G/N(G) is a free abelian group of rank n-1 and X is a complex torus, or G is a free abelian group of rank n-1. If the last case occurs, X is G-equivariant birational to the quotient of an abelian variety provided that X is a projective manifold of dimension n > 2 and is no…

On the Kobayashi pseudometric, complex automorphisms and hyperkaehler manifolds

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a compact manifold are isomorphic. We also give a proof of Kobayashi's conjecture on the vanishing of the pseudodistance for hyperkähler manifolds having Lagrangian fibrations without multiple fibers in codimension one. For a hyperbolic automorphi…

Dynamics of automorphisms on compact Kähler manifolds

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.

Criteria for the existence of equivariant fibrations on algebraic surfaces and hyperkähler manifolds and equality of automorphisms up to powers - a dynamical viewpoint

Let $X$ be a projective surface or a hyperkähler manifold and $G \le Aut(X)$. We give a necessary and sufficient condition for the existence of a non-trivial $G$-equivariant fibration on $X$. We also show that two automorphisms $g_i$ of positive entropy and polarized by the same nef divisor are the same up to powers, provided that either $X$ is not an abelian surface or the $g_i$ share at least one common periodic point. The surface case is known among experts, but we treat this case together with the hyperkähler case using the same language of hyperbolic lattice and following Ratcliffe or Ogu…

Exponential mixing for automorphisms on compact Kaehler manifolds

Let f be a holomorphic automorphism of positive entropy on a compact Kaehler surface. We show that the equilibrium measure of f is exponentially mixing. The proof uses some recent development on the pluripotential theory. The result also holds for automorphisms on compact Kaehler manifolds of higher dimension under a natural condition on their dynamical degrees.

Exponential mixing property for automorphisms of compact Kähler manifolds

Let $f$ be a holomorphic automorphism of a compact Kähler manifold. Assume moreover that $f$ admits a unique maximal dynamic degree $d_p$ with only one eigenvalue of maximal modulus. Let $μ$ be its equilibrium measure. In this paper, we prove that $μ$ is exponentially mixing for all d.s.h.\ test functions.

紧致Kähler流形上自同构的鞍型周期点的等分布

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。

Developing Maps and Engel Automorphisms

A completely nonintegrable $2$-dimensional distribution on a $4$-manifold is called an Engel structure. A $4$-manifold with an Engel structure is called an Engel manifold. The developing map for an Engel manifold is very important tool to determine the Engel structure. Montgomery used it to prove that an Engel automorphism is determined by the values on a global slice. Moreover, Montgomery constructed Engel manifolds whose automorphism group is small. In this paper, we prove that the automorphism group of an Engel manifold is embedded into the automorphism group of the Cartan prolongation of a…

Automorphisms of $\mathbb C^k$ and associated compact complex manifolds

In this paper, we first construct $k$-dimensional compact complex manifolds from automorphisms of $\mathbb{C}^k$ which admit a fixed attracting point at infinity. Then, we charactize the fundamental group as well as the universal covering of the attracting basin of this fixed point thanks to a generalization of the method described by T. Bousch in his thesis.

Chaotic holomorphic automorphisms of Stein manifolds with the volume density property

Let $X$ be a Stein manifold of dimension $n\geq 2$ satisfying the volume density property with respect to an exact holomorphic volume form. For example, $X$ could be $\mathbb{C}^n$, any connected linear algebraic group that is not reductive, the Koras-Russell cubic, or a product $Y\times\mathbb{C}$, where $Y$ is any Stein manifold with the volume density property. We prove that chaotic automorphisms are generic among volume-preserving holomorphic automorphisms of $X$. In particular, $X$ has a chaotic holomorphic automorphism. A proof for $X=\mathbb{C}^n$ may be found in work of Fornæss and Sib…

On automorphisms of blowups of projective manifolds

In this paper we mainly study the following question: For what projective manifold $X$ of dimension $\geq 3$ that any $f\in Aut(X)$ has zero topological entropy? Using some non-vanishing conditions on nef cohomology classes, we study the case where $X\rightarrow X_0$ is a finite blowup along smooth centers, here $X_0$ is a projective manifold of interest. Here we allow $X_0$ to be either one of the following manifolds: it has Picard number 1, or a Fano manifold, or it is a projective hyper-Kähler manifold. We also allow the centers of blowups to have large dimensions relative to that of $X_0$ …

Super-potentials for currents on compact Kaehler manifolds and dynamics of automorphisms

We introduce a notion of super-potential (canonical function) associated to positive closed (p,p)-currents on compact Kaehler manifolds and we develop a calculus on such currents. One of the key points in our study is the use of deformations in the space of currents. As an application, we obtain several results on the dynamics of holomorphic automorphisms: regularity and uniqueness of the Green currents. We also get the regularity, the entropy, the ergodicity and the hyperbolicity of the equilibrium measures.

紧 Kähler 流形自同构的所有阶指数混合与 CLT

Exponential mixing of all orders and CLT for automorphisms of compact K{ä}hler manifolds

本文研究上同调作用简单的紧 Kähler 流形自同构，证明其唯一最大熵测度对 Hölder 可观测量具有所有阶指数混合性，并推出 CLT，结果覆盖所有正熵紧 Kähler 曲面自同构。

Zero entropy automorphisms of compact Kähler manifolds and dynamical filtrations

We study zero entropy automorphisms of a compact Kähler manifold $X$. Our goal is to bring to light some new structures of the action on the cohomology of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbb R})$. Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations $(g^m)^* \, {\circlearrowleft} \, H^2(X, {\mathbb C})$ where $g$ is a zero entropy automorphism, in terms of ${\rm dim} \, X$ only. We also give an upper bound for the (essential) derived length $\ell_{\rm ess}(G, X)$ for every zero entropy subgroup $…

Representations of toral automorphisms

This survey gives an account of an algebraic construction of symbolic covers and representations of ergodic automorphisms of compact abelian groups, initiated by A.M. Vershik around 1992 for hyperbolic automorphisms of finite-dimensional tori. The key ingredient in this approach, which was subsequently extended to arbitrary expansive automorphisms of compact abelian groups, is the use of homoclinic points of the automorphism. Although existence and abundance of homoclinic points is intimately connected to expansiveness of the automorphism, it is nevertheless possible to extend certain aspects …

Kähler流形自同构的大熵测度的支撑集

On the support of measures of large entropy for automorphisms of Kähler manifolds

证明具有简单上同调作用的紧Kähler流形自同构的每个大熵遍历测度都支撑在Julia集上，对曲面则正熵测度支撑在Julia集上，证明依赖于Green电流收敛速度的定量估计。

Expansive Automorphisms on Locally Compact Groups

We show that any connected locally compact group which admits an expansive automorphism is nilpotent. We also show that for any locally compact group $G$, $α\in {\rm Aut}(G)$ is expansive if and only if for any $α$-invariant closed subgroup $H$ which is either compact or normal, the restriction of $α$ to $H$ is expansive and the quotient map on $G/H$ corresponding to $α$ is expansive. We get a structure theorem for locally compact groups admitting expansive automorphisms. We prove that an automorphism on a non-discrete locally compact group can not be both distal and expansive.

Topological symmetries of simply-connected four-manifolds and actions of automorphism groups of free groups

Let $M$ be a simply connected closed $4$-manifold. It is proved that any (possibly finite) compact Lie group acting effectively and homologically trivially on $M$ by homeomorphisms is an abelian group of rank at most two. As applications, let $\mathrm{Aut}(F_{n})$ be the automorphism group of the free group of rank $n.$ We prove that any group action of $\mathrm{Aut}% (F_{n})$ $(n\geq 4)$ on $M\neq S^{4}$ by homologically trivial homeomorphisms factors through $\mathbb{Z}/2.$ Moreover, any action of $% \mathrm{SL}_{n}(\mathbb{Q})$ $(n\geq 4)$ on $M\neq S^{4}$ by homeomorphisms is trivial.

On the existence of ergodic automorphisms in ergodic ${\mathbb Z} ^d$-actions on compact groups

Let $K$ be a compact metrizable group and $\Ga$ be a finitely generated group of commuting automorphisms of $K$. We show that ergodicity of $\Ga$ implies $\Ga$ contains ergodic automorphisms if center of the action, $Z(\Ga) = \{\ap \in {\rm Aut}(K) \mid \ap {\rm commutes with elements of \rm} \Ga \}$ has DCC. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which in particular, retrieves Theorems of Berend \cite{Be} and Schmidt \cite{Sc1} proved in this context.

Generic aspects of holomorphic dynamics on highly flexible complex manifolds

We prove closing lemmas for automorphisms of a Stein manifold with the density property and for endomorphisms of an Oka-Stein manifold. In the former case we need to impose a new tameness condition. It follows that hyperbolic periodic points are dense in the tame non-wandering set of a generic automorphism of a Stein manifold with the density property and in the non-wandering set of a generic endomorphism of an Oka-Stein manifold. These are the first results about holomorphic dynamics on Oka manifolds. We strengthen previous results of ours on the existence and genericity of chaotic volume-pre…

Stable manifolds for holomorphic automorphisms

We give a sufficient condition for the abstract basin of attraction of a sequence of holomorphic self-maps of balls in \mathbb{C}^{d} to be biholomorphic to \mathbb{C}^{d}. As a consequence, we get a sufficient condition for the stable manifold of a point in a compact hyperbolic invariant subset of a complex manifold to be biholomorphic to a complex Euclidean space. Our result immediately implies previous theorems obtained by Jonsson-Varolin and by Peters; in particular, we prove (without using Oseledec's theory) that the stable manifold of any point where the negative Lyapunov exponents are w…

Green currents for holmorphic automorphisms of compact Kahler manifolds

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…

具有密度性质的Stein流形的generic自同构的动力学

Dynamics of generic automorphisms of Stein manifolds with the density property

该研究针对具有密度性质的Stein流形的generic自同构开展动力学分析，刻画了其混沌最大正向不变集与Julia集的关系，并推广了全纯Kupka-Smale定理，相关结果在$\mathbb C^n$情形下也多为全新结论。

3-manifolds that admit knotted solenoids as attractors

Motivated by the study in Morse theory and Smale's work in dynamics, the following questions are studied and answered: (1) When does a 3-manifold admit an automorphism having a knotted Smale solenoid as an attractor? (2) When does a 3-manifold admit an automorphism whose non-wandering set consists of Smale solenoids? The result presents some intrinsic symmetries for a class of 3-manifolds.

每个局部紧群都是一个 II$_1$ 因子的外自同构群

Every locally compact group is the outer automorphism group of a II$_1$ factor

本文证明每个局部紧第二可数群都可实现为某个 II$_1$ 因子的外自同构群，方法是通过证明其为中心化子群，从而推广了此前仅适用于全不连通群和紧群的结果。