Dense 2-generator subsemigroups of $2\times 2$ matrices

TL;DR

This paper investigates the conditions under which subsemigroups generated by two or three elements are dense within certain matrix and transformation groups, providing explicit examples and extending results to complex matrices.

Contribution

It demonstrates the non-existence of 2-generator dense subsemigroups in some transformation groups, constructs explicit 2- and 3-generator dense subsemigroups of matrices, and extends results to complex matrices.

Findings

No 2-generator dense subsemigroups in certain real linear fractional transformations.

Explicit 2-generator dense subsemigroup of 2x2 real matrices.

Uncountably many 2-generator dense subsemigroups exist in the complex case.

Abstract

We 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. We also construct an explicit example of a 2-generator dense subsemigroup of $2\times 2$ real matrices. In the complex case, we prove the existence of uncountably many 2-generator dense subsemigroups.