Zero entropy and bounded topology

  • Gabriel P. Paternain

    University of Cambridge, United Kingdom
  • Jimmy Petean

    Guanajuato, Mexico


We study the existence of Riemannian metrics with zero topological entropy on a closed manifold with infinite fundamental group. We show that such a metric does not exist if there is a finite simply connected CW complex which maps to in such a way that the rank of the map induced in the pointed loop space homology grows exponentially. This result allows us to prove in dimensions four and five, that if admits a metric with zero entropy then its universal covering has the rational homotopy type of a finite elliptic CW complex. We conjecture that this is the case in every dimension.

Cite this article

Gabriel P. Paternain, Jimmy Petean, Zero entropy and bounded topology. Comment. Math. Helv. 81 (2006), no. 2, pp. 287–304

DOI 10.4171/CMH/53