Dynamical systems of M\"obius transformation: real, $p$-adic and complex variables

In this paper we consider function $f(x)={x+a\over bx+c}$, (where $b\ne 0$, $c\ne ab$, $x\ne -{c\over b}$) on three fields: the set of real, $p$-adic and complex numbers. We study dynamical systems generated by this function on each field separately and give some comparison remarks. For real variable case we show that the real dynamical system of the function depends on the parameters $(a,b,c)\in \mathbb R^3$. Namely, we classify the parameters to three sets and prove that: for the parameters from first class each point, for which the trajectory is well defined, is a periodic point of $f$; for the parameters from second class any trajectory (under $f$) converges to one of fixed points (there may be up to two fixed points); for the parameters from third class any trajectory is dense in $\mathbb R$. For the $p$-adic variable we give a review of known results about dynamical systems of function $f$. Then using a recently developed method we give simple new proofs of these results and prove some new ones related to trajectories which do not converge. For the complex variables we give a review of known results.