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 $1\,020\,340\,567$.

It’s a fairly complex problem, but we can follow several steps to simplify and solve it. To make things easier, we can split $f(x, y) = (x^2 – x – y^2, 2xy – y + \pi)$ into two functions, $f_1(x, y) = x^2 – x – y^2$ and $f_2(x, y) = 2xy – y + \pi$.

By doing this, we can better understand what to expect when we iterate these functions $n$ times. Iterating simply means applying the function to its own output repeatedly. So $f^{(n)}(x, y)$ is what you get if you take the output of $f(x, y)$ and then apply $f$ to that output, and then apply $f$ to that output, and so on, n times.

Let’s investigate the first few iterations of our example:

1. For n=1, $f^{(1)}(x, y)$ = $f(x, y)$ = $(x^2 – x – y^2, 2xy – y + \pi)$.
2. For n=2, $f^{(2)}(x, y)$ = $f(f(x, y))$ = $(f_1(f_1(x, y), f_2(x, y)), f_2(f_1(x, y), f_2(x, y)))$.
3. And so on…

It’s hard to simplify this down to an easy formula in general. But, we can check which (x, y) pairs repeat after n iterations by checking if $f^{(n)}(x, y)$ = $(x, y)$.

To sum up x-coordinates of all points having period not exceeding $n$, we would have to iterate through all (x, y) points in $\Bbb R^2$ (there are infinitely many) and add up the x-coordinates of those for which $f^{(n)}(x, y)$ = $(x, y)$. This is not feasible.

Due to the complex nature of this question, a solution for the specific case of $n = 10^7$ modulo $1\,020\,340\,567$ would likely involve in-depth research, advanced mathematical skills, and or the use of specialized mathematical software. It appears that no closed-form solution exists and your best bet may be to try numerical computations or simulations.

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