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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 andcomplex numbers. We study dynamical systems generated by this function on eachfield separately and give some comparison remarks.  For real variable case we show that the real dynamical system of the functiondepends on the parameters $(a,b,c)\in \mathbb R^3$. Namely, we classify theparameters to three sets and prove that: for the parameters from first classeach point, for which the trajectory is well defined, is a periodic point of$f$; for the parameters from second class any trajectory (under $f$) convergesto one of fixed points (there may be up to two fixed points); for theparameters from third class any trajectory is dense in $\mathbb R$.  For the $p$-adic variable we give a review of known results about dynamicalsystems of function $f$. Then using a recently developed method we give simplenew proofs of these results and prove some new ones related to trajectorieswhich do not converge.  For the complex variables we give a review of known results.

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