Nonexistence of codimension one Anosov flows on compact manifolds with boundary

A flow is Anosov if it exhibits contracting and expanding directions forming with the flow a continuous tangent bundle decomposition. An Anosov flow is codimension one if its contracting or expanding direction is one-dimensional. Examples of codimension one Anosov flows on compact boundaryless manifolds can be exhibited in any dimension 3. In this paper, we prove that there are no codimension one Anosov flows on compact manifolds with boundary. The proof uses an extension to flows of some results in Hirsch [On Invariant Subsets of Hyperbolic Sets, Essays on Topology and Related Topics, Memoires dédiés à Georges de Rham, 1970, pp. 126–135] related to Question 10(b) in Palis and Pugh [Fifty problems in dynamical systems, in: J. Palis, C.C. Pugh (Eds.), Dynamical Systems-Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E.C. Zeeman on his fiftieth birthday), Lecture Notes in Mathematics, vol. 468, Springer, Berlin, 1975, pp. 345–353].