论文标题
某些理性自图的周期性点和算术程度
Periodic points and arithmetic degrees of certain rational self-maps
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the second dynamical degrees distinct. We give a boundedness result about heights of its periodic points. This is motivated by a conjecture of Silverman for polynomial automorphisms of affine spaces. We also study the Kawaguchi--Silverman conjecture concerning dynamical and arithmetic degrees for certain rational self-maps in dimension two. In particular, we reduce the problem to the dynamical Mordell--Lang conjecture and verify the Kawaguchi--Silverman conjecture for some new cases. As a byproduct of the argument, we show the existence of Zariski dense orbits in these cases.
