Consider the fractional linear transformation of the real variable $t$, transformation $$f(t)=\frac{at+b}{ct+d}$$ where $a,b,c,d\in\mathbb{R}$. Define $t_{n+1}=f(t_n)$ where $t_0\in\mathbb{R}$. There are plenty of references indicating how this sequence behaves when $f$ has either two distinct fixed points $\alpha,\beta\in\mathbb{R}$ and when $f$ has one fixed point $\alpha\in\mathbb{R}$. However, I couldn't find anything that indicates the behaviour (however nasty) when $f$ has no fixed point. Is there any reference that indicates how this is analysis might proceed?
I'm actually thinking of $f$ as an automorphism of the circle (or equivalently the real projective line). I was wondering if there is a result such as $\{t_n\}$ is dense in the circle (except for certain combinations of $a,b,c,d$ that give cyclic behaviour).
Theorem: A real fractional-linear transformation without real fixed points is conjugate to a real rotation, and therefore either has finite order (in which case every orbit is finite) or infinite order (in which case each orbit is dense in the real projective line).
Proof: The eigenvalues of a fixed-point free real Möbius transformation $f$ are complex conjugates, say $\alpha \pm \beta i$. The real Möbius transformation $T(z) = (z - \alpha)/\beta$ sends $\alpha \pm \beta i$ to $\pm i$, so the real Möbius transformation $g = T \circ f \circ T^{-1}$ fixes $\pm i$, and is therefore a rotation of the Riemann sphere, which restricts to a rotation of the real projective line through an angle $\theta$. The order of $g$, and hence of $f$, if finite if and only if $\theta$ is a rational multiple of $2\pi$.
If $f$ has infinite order, the orbits of $f$ in the complex projective line are dense subsets of Apollonian circles (with two exceptions, the complex fixed points of $f$).