Infinitesimal obstructions to weakly mixing |
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Authors: | Horst Elmar Winkelnkemper |
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Affiliation: | (1) Department of Mathematics, University of Maryland, College Park, 20742 Maryland, U.S.A. |
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Abstract: | Let t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold Mn, whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d t and, for each x M, we define a smooth real function x(t) : (1 + i(t)), where the i(t) are the eigenvalues of AAT, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d 2t, restricted to v at the point x-t Mn.Among other things, we prove theTheorem (Theorem II, below). Assume v is also volume preserving and that x '(t) 0 for all x M and real t; then, if xt: M M is weakly missng for some t, it is necessary that ![lceil](/content/x626737777v6w7q6/xxlarge8968.gif) v x 0 at all x M. |
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Keywords: | Geodesic orbits weakly mixing graph of a foliation |
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