Transitivity of codimension-one Anosov actions of ℝ^k on closed manifolds

Abstract

We consider Anosov actions of ℝk, k≥2, on a closed connected orientable manifold M, of codimension one, i.e. such that the unstable foliation associated to some element of ℝk has dimension one. We prove that if the ambient manifold has dimension greater than k+2, then the action is topologically transitive. This generalizes a result of Verjovsky for codimension-one Anosov flows.

On integrable codimension one Anosov actions of $\RR^k$

“…If E ss ⊕ E uu is C 1 and if φ is volume-preserving, then φ is topologically equivalent to a suspension of a linear Anosov action of Z k on T n . More generally, it seems reasonable to conjecture, as already mentioned in [2] 1 :…”

“…The foliation F s is C 1 , hence it makes sense for a transverse affine structure to be C 1 , but not to be C 2 . In [2] we proved that among C 2 affine structures along the leaves of F uu varying continuously with the leaf, there is one and only one that is preserved by φ.…”

“…Of course, analogous statements hold for the strong and weak unstable leaves. For more details about this remark see [2]. We call a connected component of A as a chamber, by analogy with the case of Cartan actions.…”

“…Then H is a free abelian group, isomorphic to Z k , preserving g u . Moreover, the C 2 -affine structure along F uu (x) constructed in [2] provides a C 2 parametrization of g u by R, so that for this parametrization, the action of H is an action by scalar multiplications, through a morphism ρ c : H → R + * :…”

Contact Anosov actions with smooth invariant bund

“…This conjecture was latter generalized to Anosov R k actions on manifolds of dimension ≥ 3 + k. Some results in this direction were given by J. Plante (PLANTE, 1981), E. Ghys (GHYS., 1988) and T. Barbot and C. Maquera (BARBOT;MAQUERA, 2011).…”