Dense 2-generator subsemigroups of matrices

We construct explicit examples of 2-generator dense subsemigroups of GL ( 2 , R ) and SL ( 2 , R ) . In the complex case, we prove the existence of uncountably many 2-generator dense subsemigroups of GL ( 2 , C ) . We also show that the semigroup of real linear fractional transformations on a proper subinterval of the real line does not admit any 2-generator dense subsemigroups, and then we construct a 3-parameter family of examples of 3-generator dense subsemigroups.