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首页> 外文期刊>Physica, A. Statistical mechanics and its applications >NEW INTEGRABLE CASES OF A CREMONA TRANSFORMATION - A FINITE-ORDER ORBITS ANALYSIS
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NEW INTEGRABLE CASES OF A CREMONA TRANSFORMATION - A FINITE-ORDER ORBITS ANALYSIS

机译：克雷莫纳变换的新可合并案例-有限阶轨道分析

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摘要

We analyse the properties of a particular birational mapping of two variables (Cremona transformation) depending on two free parameters (epsilon and alpha), associated with the action of a discrete group of non-linear (birational) transformations on the entries of a q x q matrix. This mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations, namely taking the inverse of q x q matrices and permuting some of the entries of these matrices. It has been seen to yield weak chaos and integrability. We have found new integrable cases of this Cremona transformation, corresponding to the values of alpha=0 when epsilon=1/2,1/3,+1, besides the already known values epsilon=0 and epsilon=-1, and also arbitrary alpha when epsilon=0. For these cases, one has a foliation of the parameter space in elliptic curves. We give the equations of these elliptic curves. Based on this very example we show how one can find these integrability cases of the Cremona transformation and actually integrate it using a method based on the systematic study of the finite-order conditions of the Cremona transformation. The method is shown to be efficient and straightforward. The various integrability cases are revisited using many different representations of this very mapping (birational transformations, recursion in one variable,...). [References: 16]
机译：我们根据两个自由参数（epsilon和alpha）分析两个变量的特定双比例映射（克雷莫纳变换）的属性，这与离散的一组非线性（双比例）变换对q x q矩阵项的作用有关。此映射源自对非常简单的代数计算所获得的双值变换的分析，即对q x q矩阵求逆并置换这些矩阵的某些项。已经看到它会产生微弱的混乱和可集成性。我们发现了这种克雷莫纳变换的新可积情况，除了epsilon = 0和epsilon = -1之外，还对应于epsilon = 1 / 2,1 / 3，+ 1时alpha = 0的值，并且任意当epsilon = 0时为alpha。对于这些情况，椭圆曲线的参数空间是叶状的。我们给出这些椭圆曲线的方程。基于这个非常好的例子，我们展示了如何找到Cremona变换的这些可积性案例，并使用一种基于对Cremona变换的有限阶条件进行系统研究的方法，将其实际集成。该方法被证明是有效和直接的。使用此映射的许多不同表示形式（双向转换，一个变量的递归等），重新讨论了各种可积性情况。 [参考：16]

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