Skip to main content Accessibility help
×
Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-14T17:28:36.054Z Has data issue: false hasContentIssue false

12 - Geodesic Flows

from Part III - Ergodic Theory of Smooth and SRB Measures

Published online by Cambridge University Press:  05 May 2013

Luis Barreira
Affiliation:
Instituto Superior Técnico, Lisboa
Yakov Pesin
Affiliation:
Pennsylvania State University
Get access

Summary

For a long time geodesic flows have played an important stimulating role in the development of hyperbolic theory. In the beginning of the twentieth century, Hadamard and Morse, while studying the statistics of geodesics on surfaces of negative curvature, pointed out that the local instability of trajectories gives rise to some global properties of dynamical systems such as ergodicity and topological transitivity. The further study of geodesic flows and some objects closely related to them (e.g., frame flows) later inspired the introduction of different classes of hyperbolic dynamical systems (e.g., Anosov systems, uniformly partially hyperbolic systems, and nonuniformly hyperbolic systems preserving volume). On the other hand, geodesic flows always were a touchstone for applying new advanced methods of the general theory of dynamical systems. This, in particular, has led to some new interesting results in differential and Riemannian geometry. In this chapter we will present some of these results. While describing ergodic properties of geodesic flows in the spirit of the book, we consider the Liouville measure that is invariant under the flow (we allow other invariant measures when discussing an upper bound for the measure-theoretic entropy), and we refer the reader to the excellent survey [115] where measures of maximal entropy are studied.

Type
Chapter
Information
Nonuniform Hyperbolicity
Dynamics of Systems with Nonzero Lyapunov Exponents
, pp. 385 - 416
Publisher: Cambridge University Press
Print publication year: 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Geodesic Flows
  • Luis Barreira, Instituto Superior Técnico, Lisboa, Yakov Pesin, Pennsylvania State University
  • Book: Nonuniform Hyperbolicity
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107326026.014
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Geodesic Flows
  • Luis Barreira, Instituto Superior Técnico, Lisboa, Yakov Pesin, Pennsylvania State University
  • Book: Nonuniform Hyperbolicity
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107326026.014
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Geodesic Flows
  • Luis Barreira, Instituto Superior Técnico, Lisboa, Yakov Pesin, Pennsylvania State University
  • Book: Nonuniform Hyperbolicity
  • Online publication: 05 May 2013
  • Chapter DOI: https://doi.org/10.1017/CBO9781107326026.014
Available formats
×