# General term of an homographic sequence using auxilary sequences

Given a recursive homographic sequence : $$a_{n+1} = f(a_n)$$

where $$f$$ is an homographic function : $$x \longmapsto \dfrac{ax+b}{cx+d}$$

Where $$a,b,c,d \in \mathbb{C}$$ such that : $$ad-bc \neq 0$$

The usual way to solve this type of recurrence relation is to define a sequence $$(V_n)_{n \in \mathbb{N} }$$ , such that :

Case 1 : $$(U_n)$$ has two fixed points ($$\alpha$$,$$\beta$$)

$$\forall n \in \mathbb{N}: V_n=\frac{U_n- \alpha}{U_n - \beta}$$

Then $$(V_n)$$ is a geometric sequence with common ration $$r=\frac{c\beta +d}{c\alpha +d}$$

Note $$U_0$$ has to be different than $$\alpha$$ or $$\beta$$ otherwise $$(U_n)$$ is stationary.

Case 2 : $$(U_n)$$ only admits one fixed point ($$\alpha$$)

$$\forall n \in \mathbb{N} : V_n=\frac{1}{U_n-\alpha}$$

Then $$(V_n)$$ is an arithmetic sequence with common difference $$d=\frac{2c}{a+d}$$

Note $$U_0$$ has to be different than $$\alpha$$ otherwise $$(U_n)$$ is stationary.

My questions :

How would one come up with such a sequence $$(V_n)$$ ?

What are the fixed points role in deriving the sequence $$(V_n)$$ ?

What is the intuition behind the sequence ($$V_n$$) and its intepretation ?

Also , In one of my attemps to solve a specific homographic recurrence relation , I've found that in the case where $$f$$ admits two fixed points , the common ratio $$r$$ equals the ratio of the eigen values of the matrix $$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ Why ?

I'd like to note that i did my research on the subject but i could not find any satisfactory answer , it appears that all the documents i've consulted did not motivate the solution and rather stated it as is .

Any piece of documentation would be greatly appreciated !

As you noticed, given a homographic function (Moebius transformaion) $$f: x \longmapsto \dfrac{ax+b}{cx+d}$$ where $$a,b,c,d \in \mathbb{C}$$ such that : $$ad-bc \neq 0,$$ There are only two cases.
Case 1. The function $$f$$ has two fixed points. This case is equivalent under a suitable homography to fixed points $$0$$ and $$\infty$$ which is $$x \mapsto rx$$ for some $$r$$. In more detail, Suppose that $$f_1, f_2$$ are the two fixed points and $$s$$ is any other number. By the property of homographies, there is a unique one $$g$$ which maps $$(0, \infty, 1)$$ (in that order) to $$(f_1, f_2, s)$$. Conjugate $$f$$ by $$g$$ to get the map $$h: x \mapsto g^{-1}(f(g(x))).$$ Now notice that $$h(0) = g^{-1}(f(g(0))) = g^{-1}(f(f_1)) = g^{-1}(f_1) = 0$$ which implies that $$0$$ is a fixed point of $$h$$. Similarly for $$\infty$$.
Case 2. The function $$f$$ has one fixed point. This case is equivalent under a suitable homography to fixed point $$\infty$$ which is $$x \mapsto \delta+x$$ for some $$\delta$$.