Minimal entropy and geometric decompositions in dimension four

We show vanishing results about the infimum of the topological entropy of the geodesic flow of homogeneous smooth four-manifolds. We prove that any closed oriented geometric four-manifold has zero minimal entropy if and only if it has zero simplicial volume. We also show that if a four-manifold $M$ admits a geometric decomposition in the sense of Thurston and does not have geometric pieces modelled on hyperbolic four-space $ℍ^{4}$ , the complex hyperbolic plane $ℍ_{ℂ}^{2}$ or the product of two hyperbolic planes $ℍ^{2} \times ℍ^{2}$ then $M$ admits an $ℱ$ –structure. It follows that $M$ has zero minimal entropy and collapses with curvature bounded from below. We then analyse whether or not $M$ admits a metric whose topological entropy coincides with the minimal entropy of $M$ and provide new examples of manifolds for which the minimal entropy problem cannot be solved.

PDF Access Denied

Warning: We have not been able to recognize your IP address 47.88.87.18 as that of a subscriber to this journal. Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form.

Or, visit our subscription page for instructions on purchasing a subscription. You may also contact us at contact@msp.org or by using our contact form.

Or, you may purchase this single article for USD 29.95:

Keywords

minimal entropy, geodesic flows, geometric structures

Mathematical Subject Classification 2000

Primary: 37B40, 57M50

Secondary: 22F30, 53D25

References

Publication

Received: 21 April 2008

Revised: 5 February 2009

Accepted: 5 February 2009

Published: 25 February 2009

Authors

Pablo Suárez-Serrato
CIMAT Jalisco S/N Col. Valenciana CP: 36240 Guanajuato, Gto México.

Volume 9, issue 1 (2009)

Pablo Suárez-Serrato

Abstract