Pick $x_0 \in \mathbb{R}$ and define the following map:
$$x_n=\begin{cases} 1 & x_{n-1}=1, \\ x_{n-1}-1, & x_{n-1} \geq 2 \\ \frac{1}{x_{n-1}-1}, & x_{n-1} < 2 \end{cases}$$
It gives the following results:
For $x_0 <1$ the sequence converges to negative reciprocal golden ratio: $$x_{\infty}=1-\phi$$
For rational $x_0 \in \mathbb{Q}$ the sequence converges to 1: $$x_{\infty}=1$$
- For $x_0 =p+\sqrt{q}$ with $p,q \in \mathbb{Q}$ the sequence eventually becomes periodic. (For $x_0=\phi$ we have $x_n=\phi$ for any $n$).
- For all other cases the sequence seems to be chaotic with no clear pattern, see for example the dependence of $x_n(n)$ for the case $x_0=\pi$:
I'm interested the most in the periodic case. Let's introduce $T$ - the period of the sequence, or the number of elements in the repeating subsequence.
For simplicity, let's consider just the case of $x_0=\sqrt{M}$, with $M \in \mathbb{N}$. How to determine the period of the sequence? Is there a closed form for $T_M$?
Here's the illustration for $M=2,3,5,6,7,8,10,11,12$, the red points - the values of $x_n$, while the red line is $\sqrt{M}$:
All $x_n$ are algebraic integers with the square root part proportional to $\sqrt{M}$, it's easy to see if we consider the definition of the sequence.
For example the periodic part for $M=3$ is $$\{\frac{1+\sqrt{3}}{2}, \sqrt{3}, 1+\sqrt{3} \}$$
Any other insight in this map's properties is welcome as well.
An appendix on the geometrical interpretation of the sequence. Let:
$$x_n=\frac{a_n^2}{b_n^2}$$
Where $a_n$ - semi-major axis of an ellipse, $b_n$ - semi-minor axis of an ellipse.
At each step we draw a new elliple enclosed by the focal points of the old one. Here's an example for $a=3^{1/4}$, $b=1$:
Thus, if the sequence terminates we obtain a circle, and if it's periodic we obtain an infinite nested structure with the same pattern of $T$ different ellipses.