Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree [Formula: see text], while for the [Formula: see text] period-20 orbits the degree rises already to [Formula: see text]. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map.

中文翻译：

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二次图周期轨道的前周期与系统提取

二次映射的迭代产生多项式序列，其度数随着轨道周期的增长而爆炸。混合所有 335 个周期 12 轨道的多项式具有度数 [公式：参见文本]，而对于 [公式：参见文本] 周期 20 轨道，度数已经上升到 [公式：参见文本]。在这里，我们展示了如何使用前周期点来系统地提取精确的运动方程，一个一个，一个一个，而不需要任何迭代。精确的轨道方程为定义地图的轨道点和分岔级联的代数数塔的算术结构和嵌套属性提供了有价值的见解。