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Problem 802

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Iterated Composition

Let $\Bbb R^2$ be the set of pairs of real numbers $(x, y)$. Let $\pi = 3.14159\cdots$.

Consider the function $f$ from $\Bbb R^2$ to $\Bbb R^2$ defined by $f(x, y) = (x^2 - x - y^2, 2xy - y + \pi)$, and its $n$-th iterated composition $f^{(n)}(x, y) = f(f(\cdots f(x, y)\cdots))$. For example $f^{(3)}(x, y) = f(f(f(x, y)))$. A pair $(x, y)$ is said to have period $n$ if $n$ is the smallest positive integer such that $f^{(n)}(x, y) = (x, y)$.

Let $P(n)$ denote the sum of $x$-coordinates of all points having period not exceeding $n$. Interestingly, $P(n)$ is always an integer. For example, $P(1) = 2$, $P(2) = 2$, $P(3) = 4$.

Find $P(10^7)$ and give your answer modulo $1\ 020\ 340\ 567$.

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迭代复合函数

记$\Bbb R^2$为所有实数对$(x, y)$构成的集合。记$\pi = 3.14159\cdots$。

考虑由$\Bbb R^2$映射到$\Bbb R^2$的函数$f$，其定义为$f(x, y) = (x^2 - x - y^2, 2xy - y + \pi)$，以及其$n$次迭代复合函数$f^{(n)}(x, y) = f(f(\cdots f(x, y)\cdots))$。举例来说，$f^{(3)}(x, y) = f(f(f(x, y)))$。对于实数对$(x, y)$，若存在最小的正整数$n$使得$f^{(n)}(x, y) = (x, y)$，则称该实数对的周期为$n$。

记$P(n)$为所有周期不超过$n$的实数对的$x$轴坐标之和。有趣的是，$P(n)$始终是整数。例如，$P(1) = 2$，$P(2) = 2$，$P(3) = 4$。

求$P(10^7)$并将你的答案对$1\ 020\ 340\ 567$取余。

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