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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Small sets of homeomorphisms which control manifolds


Author: Norman Levitt
Journal: Proc. Amer. Math. Soc. 57 (1976), 173-178
MSC: Primary 57D25; Secondary 49E15
MathSciNet review: 0405450
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0405450-1
PII: S 0002-9939(1976)0405450-1
Keywords: Controllable set of vector fields, controllable set of homeomorphisms, gradient-like vector field, topologically conjugate vector fields, twisted double theorem
Article copyright: © Copyright 1976 American Mathematical Society