There are no transitive Anosov diffeomorphisms on negatively curved manifolds (Q800003)

The author gives a short proof of the following Theorem: Let M be a closed manifold that admits a Riemannian metric of strictly negative sectional curvature. Then M admits no Anosov diffeomorphism f whose nonwandering set \(\Omega\) (f) equals M. The proof uses a result of Ruelle and Sullivan about the action of Anosov diffeomorphisms on real singular homology and also the fact that the Gromov norm on real singular homology is positive under the hypotheses of the theorem. The supporting result on the Gromov norm was obtained both by Gromov and by the author in joint work with H. Inoue.