Complex dynamics in several variables

A Landau's theorem in several complex variables

In one complex variable it is well known that if we consider the family of all holomorphic functions on the unit disc that fix the origin and with first derivative equal to 1 at the origin, then there exists a constant $ρ$, independent of the functions, such that in the image of the unit disc of any of the functions of the family there is a disc of universal radius $ρ.$ This is the so celebrated Landau's theorem. Many counterexamples to an analogous result in several complex variables exist. In this paper we introduce a class of holomorphic maps for which one can get a Landau's theorem and a B…

Classification of Complex Polynomial Vector Fields in One Complex Variable

A classification of the global structure of monic and centered one-variable complex polynomial vector fields is presented.

Groups of formal diffeomorphisms in several complex variables and closed one-forms

We study groups of formal diffeomorphisms in several complex variables. For abelian, metabelian or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimensi…

Mean Li-Yorke chaos along polynomials of several variables and prime numbers

In this paper, for any given polynomial, by analyzing the limiting behavior of ergodic averages along polynomials of several variables and prime numbers, we prove that for a topology dynamical system, positive entropy implies mean Li-Yoke chaos along non-constant polynomials of several variables and prime numbers.

On classification of measurable functions of several variables

We define a normal form (called the canonical image) of an arbitrary measurable function of several variables with respect to a natural group of transformations; describe a new complete system of invariants of such a function (the system of joint distributions); and relate these notions to the matrix distribution, another invariant of measurable functions found earlier, which is a random matrix. Bibliography: 7 titles.

多元可测函数与矩阵分布的分类

Classification of measurable functions of several variables and matrix distributions

本文研究多元可测函数的矩阵（张量）分布，揭示其作为变量变换不变量与无限置换群不变概率测度的双重性质，并证明在特定条件下二元函数的矩阵分布构成完备不变量。