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Problem 802

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Iterated Composition

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

Consider the function $f$ from ${\mathbb{R}}^2$ to ${\mathbb{R}}^2$ 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 $1020340567$.

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迭代复合函数

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

考虑由${\mathbb{R}}^2$映射到${\mathbb{R}}^2$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$1020340567$取余。

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