Small sets of homeomorphisms which control manifolds

Abstract:

Let ${N^n}$ be a smooth connected paracompact manifold without boundary. A set $D$ of self-homeomorphisms of ${M^n}$ to itself is called controllable iff the semigroup generated by $D$ acts transitively on ${M^n}$. Theorem A. There is a complete vector field $X$ on ${M^n}$ and a self-homeomorphism $H$ so that the set $D$ consisting of $H,{H^{ - 1}}$ and ${X_t},t \in {\mathbf {R}}$, is controllable. Theorem B. Let $n \ne 4$ and let ${M^n}$ be compact and orientable. If $n$ is even, $\geqslant 6$, let ${M^n}$ be simply connected. If $n \equiv 0(4)$, let signature ${M^n} = 0$. Then there is a vector field $X$ on ${M^n}$ and a self-homeomorphism $H$ so that the set consisting of $H,{H^{ - 1}}$ and ${X_t},t \geqslant 0$, is controllable.