Properties of a pair of functions which generate a dense subsemigroup of \(S(X)\) (Q1905978)

Let \(X\) be a locally compact Hausdorff space. Denote by \(S(X)\) the semigroup of all continuous mappings from \(X\) into itself with the compact-open topology. It is proved that if \(X\) is a compact subspace of \(\mathbb{R}^n\) (\(\mathbb{R}\) is the set of all reals with usual topology) and if there exist two maps in \(S(X)\) generating a dense subsemigroup of \(S(X)\) then exactly one of the following possibilities occurs: (1) one map is a homeomorphism from \(X\) properly into \(X\) and the other is surjective but not injective; (2) one map is a homeomorphism from \(X\) to \(X\) and the other is neither surjective nor injective. -- This theorem is a consequence of a more general statement. Assume that \(f_1,f_2,\dots, f_n\in S(X)\) so that the subsemigroup of \(S(X)\) generated by \(f_1, f_2,\dots, f_n\) is dense in \(S(X)\). Then a certain \(f_i\) is injective. If \(X\) is compact then a certain \(f_j\) is not surjective. If the range of any continuous map of \(X\) into itself is not a proper dense subset of \(X\) then a certain \(f_k\) is surjective. If \(X\) is a subset of \(\mathbb{R}^n\) then a certain \(f_l\) is not injective.