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2003, Volume 9, Issue 2: 263-280. Doi: 10.3934/dcds.2003.9.263
This issue Previous Article On the stability of some properly--degenerate Hamiltonian systems with two degrees of freedom Next Article A remark on reaction-diffusion equations in unbounded domains

Pointwise dimensions for Poincaré recurrences associated with maps and special flows

  • 1.

    IICO-UASLP, A. Obregón 64, 78000 San Luis Postosí, SLP

  • 2.

    Centre de Physique Théorique, CNRS-Ecole polytechnique, UMR 7644, F-91128 Palaiseau Cedex

  • 3.

    CNRS-LAMFA, Université de Picardie Jules Verne, 80039 Amiens

Received:  October 31, 2001
Revised:  April 30, 2002
Published:  February 2003
Abstract Related Papers Cited by
    Abstract
  • We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.
    Mathematics Subject Classification: 37C45, 37B20, 37A17.

    Citation:
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