Semigroups of real functions with dense orbits

Let ${\mathcal{F}}_I=\{f:I\to I\mathrm{\mid }f(x)=(Ax+B)\mathrm{/}(Cx+D);AD-BC\mathrm{\ne }0\}$, where $I$ is an interval. For $x\in I$, let ${\mathrm{\Omega }}_x$ be the orbit of $x$ under the action of the semigroup of functions generated by $f,g\in {\mathcal{F}}_I$. Our main result in this paper is to describe all $f,g\in {\mathcal{F}}_I$ such that ${\mathrm{\Omega }}_x$ is dense in $I$ for all $x$.