Dynamics of regular birational maps in P^k

Energy and Invariant Measures for Birational Surface Maps

When a birational surface map is expanding on cohomology there is a canonical way to associate positive closed currents to the map and its inverse. In this paper we use a version of Dirichlet energy to construct the wedge product of these two currents under a very weak additional condition on the map. We show that the resulting measure is invariant and mixing, and we establish that its Lyapunov exponents are finite and non-zero.

$\boldsymbol{\text{mathbb{P}}^k}$上一般双有理映射的所有阶指数混合与中心极限定理

Exponential mixing of all orders and CLT for generic birational maps of $\mathbb{P}^k$

针对$\boldsymbol{\text{mathbb{P}}^k}$上一般双有理映射的最大熵测度，推广了Hénon映射下的所有阶指数混合及Hölder可观测量CLT结论，通过引入合适的检验函数空间解决了不定集导致的Hölder映射迭代不稳定问题。

Dynamics of a two parameter family of plane birational maps: maximal entropy

We consider the real dynamics of a two parameter family of plane birational maps, focusing especially on an open subset of parameter space on which the real and complex dynamics are in close agreement. On the complex side, we find a rational complex surface to which the maps extend in a well-defined fashion and then calculate the induced action on the Picard group of the surface. On the real side, we use the critical sets of the maps to produce a combinatorial model for the dynamics. By comparing the two points of view, we are able to code points in the real non-wandering set, describe the beh…

Invariant Fibrations for some Birational Maps of C^2

In this article we extract and study the zero entropy subfamilies of a certain family of birational maps of the plane. We find these zero entropy mappings and give the invariant fibrations associated to them.

随机的birational映射乘积：曲线预像的等分布

Random products of birational maps: Equidistribution of preimages of curves

本文研究了随机birational变换乘积对平面中曲线预像的等分布性质，基于Diller和Roeder最近的工作，探讨了Cremona群在所有平面模型的当前空间逆极限上的作用，并证明了随机有限支持行走的等分布性。

On the complex dynamics of birational surface maps defined over number fields

We show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford-Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well-behaved. We also show that there is a well-defined canonical height function.

Dynamical Classification of a Family of Birational Maps of C^2 via Algebraic Entropy

This work dynamically classifies a 9-parametric family of birational maps f : C2 -> C2. From the sequence of the degrees dn of the iterates of f, we find the dynamical degree delta(f) of f. We identify when dn grows periodically, linearly, quadratically or exponentially. The considered family includes the birational maps studied by Bedford and Kim in [4] as one of its subfamilies.

Zero Entropy for Some Birational Maps of C^2

This work deals with a special case of family of birational maps f : C2 -> C2 dynamically classified in [9]. In this work we study the zero entropy sub families of f. The sequence of degrees dn associated to the iterates of f is found to grow periodically, linearly, quadratically or exponentially. Explicit invariant fibrations for zero entropy families and all the integrable and periodic mappings inside the family f are given.

Lie Symmetries of Birational Maps Preserving Genus 0 Fibrations

We prove that any planar birational integrable map, which preserves a fibration given by genus $0$ curves has a Lie symmetry and some associated invariant measures. The obtained results allow to study in a systematic way the global dynamics of these maps. Using this approach, the dynamics of several maps is described. In particular we are able to give, for particular examples, the explicit expression of the rotation number function, and the set of periods of the considered maps.

The ordinal of dynamical degrees of birational maps of the projective plane

We show that the ordinal of the dynamical degrees of all complex birational maps of the projective plane is $ω^ω$.

Periodic points of birational maps on projective surfaces

We classify birational maps of projective smooth surfaces whose non-critical periodic points are Zariski dense. In particular, we show that if the first dynamical degree is greater than one, then the periodic points are Zariski dense.

On regularizable birational maps

Bedford asked if there exists a birational self map $f$ of the complex projective plane such that for any automorphism $A$ of the complex projective plane $A\circ f$ is not conjugate to an automorphism. Blanc gave such a $f$ of degree $6$ and asked if there exists an example of smaller degree. In this article we give an example of degree $5$.

Real dynamics of a family of plane birational maps: trapping regions and entropy zero

Continuing work begun in a previous paper, we study the real dynamics of a family of plane birational maps. This time we consider a parameter range where the real and complex dynamics are different. We show in particular that for one parameter value, the real dynamics are transient even though the map is not integrable.

Degree growth of matrix inversion: birational maps of symmetric, cyclic matrices

We consider two (densely defined) involutions on the space of $q\times q$ matrices; $I(x_{ij})$ is the matrix inverse of $(x_{ij})$, and $J(x_{ij})$ is the matrix whose $ij$th entry is the reciprocal $x_{ij}^{-1}$. Let $K=I\circ J$. The set ${\cal SC}_q$ of symmetric, cyclic matrices is invariant under $K$. In this paper, we determine the degrees of the iterates $K^n=K\circ...\circ K$ restricted to ${\cal SC}_q$.

Degree growth of polynomial automorphisms and birational maps: some examples

We provide the existence of new degree growths in the context of polynomial automorphisms of $\mathbb{C}^k$: if $k$ is an integer $\geq 3$, then for any $\ell\leq \left[\frac{k-1}{2}\right]$ there exist polynomial automorphisms $f$ of $\mathbb{C}^k$ such that $\mathrm{deg}\, f^n\sim n^\ell$. We also give counter-examples in dimension $k\geq 3$ to some classical properties satisfied by polynomial automorphisms of $\mathbb{C}^2$. We provide the existence of new degree growths in the context of birational maps of $\mathbb{P}^k_\mathbb{C}$: assume $k\geq 3$; forall $0\leq\ell\leq k$ there exist bi…

Dynamics of the birational maps arising from $F_0$ and $dP_3$ quivers

The dynamics of the maps associated to $F_0$ and $dP_3$ quivers is studied in detail. We show that the corresponding reduced symplectic maps are conjugate to globally periodic maps by providing explicit conjugations. The dynamics in $\Rb^N_+$ of the original maps is obtained by lifting the dynamics of these globally periodic maps and the solution of the discrete dynamical systems generated by each map is given. A better understanding of the dynamics is achieved by considering first integrals. The relationship between the complete integrability of the globally periodic maps and the dynamics of …

Invariant curves for birational surface maps

We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and number of irreducible components of the curve. In the case of an invariant curve with genus equal to one, we show that there is an associated invariant meromorphic two form.

Transformations birationnelles de petit degré

Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals with these quadratic maps. We establish algebraic properties such as the classification of one parameter groups of quadratic birational maps or the smoothness of the set of quadratic birational maps in the set of rational maps. We prove that a finite number of generic quadratic birational maps generates a free group. We show that if f is a quadratic birational …

Complex Dynamics of birational maps of $\mathbb{P}^k$ defined over a number field

Jonsson and Reschke showed that birational selfmaps on projective surface defined over a number field satisfy the energy condition of Bedford and Diller so their ergodic properties are very well understood. Under suitable hypotheses on the indeterminacy loci, we extend that result to birational maps $\mathbb{P}^k\dashrightarrow\mathbb{P}^k$, $k\geq2$, defined over a number field, showing that they satisfy a similar energy condition introduced by De Thélin and the second author. As a consequence, we can construct for such maps their Green measure and deduce several important ergodic consequence…

具有超越数动力学度的双有理映射

Birational maps with transcendental dynamical degree

本文构造了 $\mathbb{P}^d (d \geq 3)$ 上动力学度为超越数的双有理自映射，反驳了 Bellon 和 Viallet 的猜想，证明结合了代数动力学与丢番图逼近技术。

Algebraic Entropy of Birational Maps with Invariant Curves

The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this letter, two complementary proofs are given that the foliation of the space by invariant curves implies that the algebraic entropy is zero.

Degree complexity of birational maps related to matrix inversion: Symmetric case

For $q\geq 3$, we let $\mathcal{S}_q$ denote the projectivization of the set of symmetric $q\times q$ matrices with coefficients in $\mathbb{C}$. We let $I(x)=(x_{i,j})^{-1}$ denote the matrix inversion, and we let $J(x)=(x_{i,j}^{-1})$ be the matrix whose entries are the reciprocals of the entries of $x$. We let $K|\mathcal{S}_q=I\circ J:\mathcal{S}_q\rightarrow \mathcal{S}_q$ denote the restriction of the composition $I\circ J$ to $\mathcal{S}_q$. This is a birational map whose properties have attracted some attention in statistical mechanics. In this paper we compute the degree complexity o…

General solution of functional equations defined by generic linear-fractional mappings F_1: C^N \to C^N and by generic maps birationally equivalent to F_1

We consider a system of birational functional equations (BFEs) (or finite-difference equations at w=m \in Z) for functions y(w) of the form: y(w+1)=F_n(y(w)), y(w):C \to C^N, n=deg(F_n(y)), F_n \in (\bf Bir}(C^N), where the map F_n is a given birational one of the group of all automorphisms of C^N \to C^N. The relation of the BFEs with ordinary differential equations is discussed. We present a general solution of the above BFEs for n=1,\forall N and of the ones with the map F_n birationally equivalent to F_1: F_n\equiv V\comp F_1\comp V^{-1}, \forall V \in (\bf Bir}(C^N).

Degree growth of birational maps of the plane

This article studies the sequence of iterative degrees of a birational map of the plane. This sequence is known either to be bounded or to have a linear, quadratic or exponential growth. The classification elements of infinite order with a bounded sequence of degrees is achieved, the case of elements of finite order being already known. The coefficients of the linear and quadratic growth are then described, and related to geometrical properties of the map. The dynamical number of base-points is also studied. Applications of our results are the description of embeddings of the Baumslag-Solitar …

Dynamics of (Pseudo) Automorphisms of 3-space: Periodicity versus positive entropy

We study the iterative behavior of the family of 3-step linear fractional recurrences and the family of birational maps they define. We determine all the possible periodicities within this family or, equivalently, the birational maps of finite order. This family also contains pseudo-automorphisms of infinite order. One such family consists of completely integrable maps, and another family consists of maps of positive entropy. Both of these families have invariant families of $K3$ surfaces.

A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a birational self-map $ϕ$ of dynamical degree $1$, we expect that either there exists a non-constant rational function $f:X\dashrightarrow \mathbb{P}^1$ such that $f\circ ϕ=f$, or there exists a proper subvariety $Y\subset X$ with the property that for any invariant proper subvariety $Z\subset X$, we have that $Z\subseteq Y$. We prove our conjecture for automorph…

Remarks on the degree growth of birational transformations

We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the affine space and examples of degree sequences are displayed. We also show that the set of all degree sequences of rational maps is countable; this generalizes a result of Bonifant and Fornaess.

源于簇代数突变的平面双有理映射的复动力学视角

Complex dynamics perspective for birational maps of the plane arising from cluster algebra mutations

研究源自簇代数理论和可积系统的平面双有理映射，用全纯动力学方法计算其动力学次数，证明许多映射无守恒量和不变纤维化，回答相关问题，并应用遍历理论结果产生具有正熵和正李雅普诺夫指数的不变测度。

基于关于吹胀的多项式指标的双有理映射的动力次数 II. 三维示例

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups II. 3D examples

本文提出一种利用多项式指标与拉回因式分解计算双有理映射迭代次数的新方法，通过三维示例验证其灵活性，无需构造代数稳定提升，优于传统皮卡群方法。

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups I. General theory and 2D examples

In this paper we address the problem of computing $\text{deg}(f^n)$, the degrees of iterates of a birational map $f:\mathbb{P}^N\rightarrow\mathbb{P}^N$. For this goal, we develop a method based on two main ingredients: the factorization of a polynomial under pull-back of $f$, based on local indices of a polynomial associated to blow-ups used to resolve the contraction of hypersurfaces by $f$, and the propagation of these indices along orbits of $f$. For maps admitting algebraically stable modifications $f_X:X\rightarrow X$, where $X$ is a variety obtained from $\mathbb P^N$ by a finite number…

基于极化的双有理映射及其对测度与积分的保持

Birational maps from polarization and the preservation of measure and integrals

本文利用极化方法构造双有理离散化，将次数不超过4、任意维数的二阶Hamiltonian系统推广为保持能量、不变测度和相空间维数的格式。

光滑仿射二次三维簇的一些有趣双有理态射

Some interesting birational morphisms of smooth affine quadric $3$-folds

本文研究复光滑仿射二次三维簇上一族多项式诱导的双有理映射，揭示其上同调双曲性、非代数整数的第二动力学度及周期点增长与代数熵的差异，并结合 Bertini 实验验证。

Dynamics of birational plane mappings. The Arnold complexity difference equation

We consider a dynamics of a generic birational plane map Φ_n: CP^2 \to CP^2, CP^2 -image of the birational mapping (inverse map is also rational)F_n : C^2 \to C^2 and its such important characteristic as the Arnold complexity C_A(k), which is proportional d(k)=deg(Φ_n^k)- a degree of k-iteration of the map Φ_n, on the basis on algebraic-geometrical properties of such maps. Additional importance of this characteristic follows from the Veselov conjecture about the polynomial boundedness of the growth of d(k) for integrable dynamical systems with a discrete time defined by birational plane maps. …

周期一与周期二乘子的有理映射的通用重构

Generic reconstruction of rational maps from multipliers of periods one and two

本文证明对任意次数d≥2且特征非2的域，模空间上由周期一、二乘子构成的谱态射双有理于其像闭包，从而在代数闭域上通用单射，证实了Ji-Xie猜想，方法涉及正规形、非阿基米德退化与对不变量重构。

关于有理曲面映射的稳定化

On the Stabilisation of Rational Surface Maps

研究有理曲面映射通过双有理共轭实现代数稳定化的条件与方法，证明了极小共轭与图提升构造的存在性，并指出仅靠吹胀无法实现所有稳定共轭。

某些有理自映射的周期点与算术度

Periodic points and arithmetic degrees of certain rational self-maps

本文研究代数数域上的上同调双曲双有理自映射，给出其周期点高度的有界性结果，还将二维有理自映射的Kawaguchi--Silverman猜想归约为动力Mordell--Lang猜想，验证了若干新情形并证明对应Zariski稠密轨道存在。