Topologically transitive semigroup actions of real linear fractional transformations (Q973988)

Let \(I\) be a proper closed subinterval of the real line \(\mathbb{R}\). Denote by \(\mathcal{F}_I\) the set of all linear fractional transformations \(f:I \rightarrow I\), \(f(x)=(Ax+B)/(Cx+D)\), \(AD-BC \neq 0\). This paper describes all pairs \(f,g \in \mathcal{F}_I\) such that the action of the semigroup \(\langle f,g\rangle\) generated by \(f\) and \(g\) is topologically transitive on \(I\). According to the main characterization, if one orbit is dense, then all orbits are dense, except possibly those of the boundary points. In particular, if \(I\) is a proper subinterval, the semigroup action generated by a pair of maps in \(\mathcal{F}_I\) is topologically transitive if and only if it is hypercyclic. However, hypercyclicity does not imply topological transitivity for semigroup actions generated by pairs of affine functions on \(\mathbb{R}\). The author further shows that the semigroup action \(\langle f,g\rangle\) is never weakly topologically mixing if \(I\) is a proper subinterval. He also characterizes the affine pairs in \(\mathcal{F}_{\mathbb{R}}\) generating a weakly topologically mixing semigroup. Finally, some open questions are stated.