Let Y = ( f, g, h): R 3 → R 3 be a C 2 map and let Spec(Y ) denote the set of eigenvalues of the derivative DY p , when p varies in R 3 . We begin proving that if, for some > 0, Spec(Y ) ∩ (− , ) = ∅, then the foliation F (k), with k ∈ { f, g, h}, made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek's Jacobian Conjecture for polynomial maps of R n .
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.