Lower bounds for the topological entropy

Pages: 555 - 565, Volume 12, Issue 3, March 2005

doi:10.3934/dcds.2005.12.555       Abstract        Full Text (232.6K)

Katrin Gelfert - Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (email)

Abstract: We establish lower bounds for the topological entropy expressed in terms of the exponential growth rate of $k$-volumes. This approach provides the sharpest possible bounds when no further geometric information is available. In particular, our methods apply to (partially) volume-expanding dynamics with not necessarily compact phase space, including a large class of geodesic flows. As an application, we conclude that the topological entropy of these systems is positive.

Keywords:  Topological entropy, Hausdorff dimension, geodesic flow.
Mathematics Subject Classification:  37C45, 37D40.

Received: October 2003;      Revised: August 2004;      Published: December 2004.