Foliations and noncompact transformation groups
Morris W. Hirsch · Bulletin of the American Mathematical Society · 1970
Introduction.Let G be a Lie group and M a compact C 00 manifold.In [2] Anosov actions of G on M are defined and proved to be structurally stable.In this announcement we are concerned with the foliation ^ of M defined by the orbits of G.Under the assumption that G is connected, # is C 1 stable (3).If G is connected and nilpotent, G has a compact orbit (4).If G is merely solvable, however, there may be no compact orbit.In fact it can happen that no foliation C° close to 9r has a compact leaf (8).Upper bounds for the number of compact orbits of given type are found (9).In (7) we discuss the intersection of certain nilpotent subgroups of a Lie group S with conjugates of a uniform discrete subgroup of S.Hyperbolic automorphisms of foliations.A k-foliation ff of M is a function assigning to each x £ M the image & x of a C 2 injective immersion V X ->M of a connected ^-dimensional manifold V x .We require that the leaf $ x contain x, and that the function TïïlM->Gk(M) assigning to x(E.M the tangent plane to V x at x be C 1 ; here Gk(M) is the manifold of fe-planes tangent to M. Equivalently, T$ is a completely integrable C 1 field of fe-planes, and SF» is the maximal integral submanifold through x.Thus j^}^^ is a partition of M. The set Fk(M) of all ^-foliations of M inherits the C° and C 1 topologies from the set of C 1 maps M->Gk(M).We also use TS to denote the bundle of fe-planes tangent to the leaves.If 9S gGF&(ikf), a homeomorphism h:$->2 is a homeomorphism of M taking each leaf of # onto a leaf of g.We call ^ C 1 stable if it has a C 1 neighborhood N(ZFk(M) of foliations homeomorphic to 3 r .An automorphism g of # is a C 1 diffeomorphism of ikf which is a homeomorphism 3 r -»9 r .We call g hyperbolic if there exists a splitting TM = E+@E-®T$ invariant under Tg, and such that the following condition holds.For some (and hence any) Riemannian metric on M there exist constants 0<X<1