Abstract
In this note we prove that for equilibrium states of axiom A systems having positive dimension the time τ B (x) needed for a typical point x to enter for the first time in a typical ball B with radius r behaves for small r as τ B (x)∼ r −d where d is the local dimension of the invariant measure at the center of the ball. A similar relation is proved for a full measure set of interval exchanges. Some applications to Birkoff averages of unbounded (and not L 1) functions are shown.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Barreira and B. Saussol, Hausdorff dimension of measures via Poincaré recurrence. Commun. Math. Phys. 219:443–463 (2001).
C. Bonanno, S. Isola, and S. Galatolo, Recurrence and algorithmic information. Nonlinearity 17(3):1057–1074 (2004).
M. D. Boshernitzan, Quantitative recurrence results. Invent. Math. 113:617–631 (1993).
M. D. Boshernitzan, A condition for minimal interval exchange maps to be uniquely ergodic. Duke Math. J. 52(3):723–752 (1985).
H. Bruin, B. Saussol, S. Troubetzkoy, and S. Vaienti, Return time statistics via inducing. Ergodic Theory Dynam. Sys. 23(4):991–1013 (2003).
R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lect. Notes Math. vol. 470, Springer-Verlag, 1975.
J. R. Chazottes and S. Galatolo, Using Hitting & Return times to study strange attractors (work in preparation).
T. Carletti and S. Galatolo, Numerical Estimates of dimension by recurrence and waiting time. Physica A (in print) 81–587.
S. Galatolo, Dimension via waiting time and recurrence. Math. Res. Lett. 12(3):377–386 (2005).
S. Galatolo, D. H. Kim, and K. Koh Park, Recurrence time for some infinite ergodic systems (work in preparation).
S. Gratrix and J. N. Elgin, Pointwise Dimensions of the Lorenz Attractor. Phys. Rev. Lett. 92:014101 (2004).
T. C. Halsey and M. H. Jensen, Hurricanes and butterflies. Nature 428:127–128 (2004).
D. H. Kim and B. K. Seo, The waiting time for irrational rotations. Nonlinearity 16(5):1861–1868 (2003).
C. Kim and D. H. Kim, On the law of logarithm of the recurrence time. Discrete Contin. Dyn. Syst. 10(3):5 (2004).
M. H. Jensen, L. P. Kadanoff, A. Libchaber, I. Procaccia, and J. Stavans, Global universality at the onset of chaos: Results of a forced Rayleigh-Bénard experiment. Phys. Rev. Lett. 55(25):2798–2801.
Y. Lacroix, N. Haydn, and S. Vaienti, Hitting and return times in ergodic dynamical systems. (to appear in Ann. Probab.).
Y. Pesin, Dimension theory in dynamical systems. Chicago lectures in Mathematics (1997).
B. Saussol, S. Troubetzkoy, and S. Vaienti, Recurrence, dimensions and Lyapunov exponents. J. Stat. Phys. 106:623–634 (2002).
B. Saussol, Recurrence rate in rapidly mixing dynamical systems. preprint: math.DS/0412211 (to appear on Disc. Cont. Dyn. Sys.).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Galatolo, S. Hitting Time and Dimension in Axiom A Systems, Generic Interval Exchanges and an Application to Birkoff Sums. J Stat Phys 123, 111–124 (2006). https://doi.org/10.1007/s10955-006-9041-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-006-9041-y