双有理平面映射的动力学。Arnold复杂性差分方程
Dynamics of birational plane mappings. The Arnold complexity difference equation
摘要：
本文考虑一般双有理平面映射Φ_n的动力学：CP^2 \to CP^2，CP^2 -双有理映射的象（逆映射也是有理的）F_n：C^2 \to C^2和它的Arnold复杂度C_A（k）等重要特征，即d（k）=deg（Φ_n^k）-映射Φ_n的k次迭代，基于这种映射的代数几何性质。这个特征的另一个重要性来自于Veselov猜想，关于由双有理平面映射定义的离散时间的可积动力系统的d（k）增长的多项式有界性。得到了d（k）的整系数自治线性差分方程。该方程完全由σ_1非负整数m_1，.，m_σ_1由以下关系式确定：Φ_n^{-m_i}（O_α_i）=O^（-1）_β_i，i \in（1，2，.，σ_1），其中Φ_n^{-m_i}是逆映射的m_i-迭代，O_α_i，O^（-1）_β_i，α_i，β_i \in（1，2，.，σ）是正映射和逆映射的不定点，σ_1 \leq σ和σ是Φ_n，Φ_n^{-1}的不定点的个数。如果σ_1等于零，则d（k）=n^k，否则d（k）的增长完全由与d（k）的差分方程相关联的久期方程的根谱定义。Veselov猜想对应于由模1的值组成的根谱。作者并不认为读者已经熟悉CP ^2中的代数几何（AG）和动力系统理论（DST）或函数方程，因为在本文中给出了所有所需的AG或DST概念和定理的定义。
英文摘要：
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. The autonomous linear difference equation with integer coefficients for d(k) is obtained. This equation is fully defined by σ_1 nonnegative integers m_1,..., m_σ_1 that are determined by relations: Φ_n^{-m_i}(O_α_i)=O^(-1)_β_i, i\in(1,2,...,σ_1), where Φ_n^{-m_i} is m_i-iteration of inverse map, O_α_i, O^(-1)_β_i, α_i, β_i \in (1,2,...,σ), are indeterminacy points of the direct and inverse maps, σ_1\leqσand σis a number of indeterminacy points of Φ_n,Φ_n^{-1}. If σ_1 is equal to zero that d(k)=n^k, otherwise the growth of d(k) is fully defined by a root spectrum of the secular equation associated with the difference equation for d(k). The Veselov conjecture corresponds to the root spectrum consisting of values being equal to modulo one. The author doesn't suppose that the reader has acquaintance with the algebraic geometry (AG) in CP^2 and the dynamical systems theory (DST) or the functional equations since in the paper there are given all needed definitions of used concepts of AG or DST and theorems.
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