On the Degree Growth of Birational Mappings in Higher Dimension

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.

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 …

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).

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。

双有理动力系统的度增长、轨道图与函子性

Degree growth, orbit graphs, and functoriality for birational dynamical systems

提出基于除子的计数方法，通过轨道图和除子拉回的函子性失效导出度增长线性差分系统，适用于任意维双有理动力系统。

Dynamical degrees of birational transformations of projective surfaces

The dynamical degree $λ(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational transformations of projective surfaces, and the relationship between the value of $λ(f)$ and the structure of the conjugacy class of $f$. For instance, the set of all dynamical degrees of birational transformations of the complex projective plane is a closed and well ordered set of algebraic numbers.

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. …

Degree growth of meromorphic surface maps

We study the degree growth of iterates of meromorphic selfmaps of compact Kahler surfaces. Using cohomology classes on the Riemann-Zariski space we show that the degrees grow similarly to those of mappings that are algebraically stable on some birational model.

Dynamics of meromorphic maps with small topological degree II: Energy and invariant measure

We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact Kähler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and admits a natural geometric description. Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\mathbf{C}^2$. They are new even in the birational case. We also exhibit families of mappings where our assumptions are generically satisfied and show that if counte…

通过奇点约束和次数增长的脱自治化

Deautonomisation by singularity confinement and degree growth

本文解释了平面双有理映射的次数增长与通过奇点约束进行脱自治化的联系，通过初始条件空间在系数演化与Picard格诱导动力学之间建立桥梁。

Degree Complexity of Matrix Inversion

For a q by q matrix x=(x_{i,j}) we let J(x)=(x_{i,j}^{-1}) be the Hadamard inverse, which takes the reciprocal of the elements of x . We let I(x)=(x_{i,j})^{-1} denote the matrix inverse, and we define K=I\circ J to be the birational map obtained from the composition of these two involutions. We consider the iterates K^n=K\circ...\circ K and determine degree complexity of K, which is the exponential rate of degree growth of the degrees of the iterates.

Entropy, volume growth and SRB measures for Banach space mappings

We consider $C^2$ Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volum…

Phragmén-Lindelöf 原理与拟正则映射的 Julia 极限方向

Phragmén-Lindelöf Principles and Julia Limiting Directions of Quasiregular Mappings

本文研究超越型拟正则映射的 Julia 极限方向集结构，利用允许 $O(\log |x|)$ 边界增长的新型 Phragmén-Lindelöf 原理，建立其与维数、伸缩率和增长阶的关系。

Horseshoes for $\mathcal{C}^{1+α}$ mappings with hyperbolic measures

We present here a construction of horseshoes for any $\mathcal{C}^{1+α}$ mapping $f$ preserving an ergodic hyperbolic measure $μ$ with $h_μ(f)>0$ and then deduce that the exponential growth rate of the number of periodic points for any $\mathcal{C}^{1+α}$ mapping $f$ is greater than or equal to $h_μ(f)$. We also prove that the exponential growth rate of the number of hyperbolic periodic points is equal to the hyperbolic entropy. The hyperbolic entropy means the entropy resulting from hyperbolic measures.

一类带拟周期力的仿射Anosov映射的不变环面

Invariant tori for a class of affined Anosov mappings with quasi-periodic forces

本文证明对一类带拟周期力的仿射Anosov映射，存在唯一仅依赖于系统的正整数m，使度为m的不变环面数量的指数增长率等于拓扑熵。

映射类群的子群的动力学

Dynamics of subgroups of mapping class groups

本文研究亏格不小于2的闭可定向曲面的映射类群中，Bowen-Margulis测度有限的非初等子群，给出了其关于Teichmüller度量的渐近增长公式。