Topologically transitive semigroup actions of real linear fractional transformations

Let F I = { f : I → I | f ( x ) = ( A x + B ) / ( C x + D ) ; A D − B C ≠ 0 } , where I is a proper closed subinterval of R . Our main goal in this paper is to describe all f , g ∈ F I such that the action of the semigroup generated by f and g is topologically transitive on I. We also prove that this action is never weakly topologically mixing. Finally, we describe all pairs of affine functions on R whose generated semigroup action is weakly topologically mixing.