Locally free affine group actions

Plante, J. F.

doi:10.1090/S0002-9947-1980-0567090-9

Locally free affine group actions
HTML articles powered by AMS MathViewer

by J. F. Plante PDF

Trans. Amer. Math. Soc. 259 (1980), 449-456 Request permission

Abstract:

Differentiable actions by the nonabelian 2-dimensional Lie group on compact manifolds are considered. When the action is locally free and the orbits have codimension one it is shown that there are at most finitely many minimal sets each containing a countably infinite number of cylindrical orbits. Examples are given to show that various codimension, differentiability, and minimality restrictions are necessary.

References

G. Chatelet, H. Rosenberg, and D. Weil, A classification of the topological types of $\textbf {R}^{2}$-actions on closed orientable $3$-manifolds, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 261–272. MR 346809, DOI 10.1007/BF02684372

On exceptional minimal sets

Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
Brian Marcus, Ergodic properties of horocycle flows for surfaces of negative curvature, Ann. of Math. (2) 105 (1977), no. 1, 81–105. MR 458496, DOI 10.2307/1971026
Joseph F. Plante, Diffeomorphisms with invariant line bundles, Invent. Math. 13 (1971), 325–334. MR 307263, DOI 10.1007/BF01406082
Joseph F. Plante, Anosov flows, Amer. J. Math. 94 (1972), 729–754. MR 377930, DOI 10.2307/2373755
J. F. Plante, Foliations with measure preserving holonomy, Ann. of Math. (2) 102 (1975), no. 2, 327–361. MR 391125, DOI 10.2307/1971034
V. A. Rohlin, On the fundamental ideas of measure theory, Mat. Sbornik N.S. 25(67) (1949), 107–150 (Russian). MR 0030584
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1