Multifractals via recurrence times ?
arXiv:nlin/0409017v1 [nlin.CD] 8 Sep 2004
Jean-Ren´e Chazottes 1 , Stefano Galatolo
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This letter echoes the article by T.C. Halsey and M.H. Jensen published in this journal [5]. To say it in a nutshell, the authors evoke various methods use to determine multifractal properties of strange attractors of dynamical systems, in particular from experimental data. They draw their attention on a method based on recurrence times used by S. Gratrix and J.N. Elgin [4] to estimate the pointwise dimensions of the Lorenz attractor. The aim of this letter is to point out that there is an important mathematical literature establishing positive but also negative results about the use of recurrence times to compute multifractal characteristics of strange attractors, e.g. [1] (to cite only one). This literature is not cited in [5] neither in [4]. In view of the use of recurrence times techniques for the experimental demonstration of fractal properties of a variety of natural systems, these negative results cannot overlooked. Recurrence times are simply defined by considering a point on the attractor and asking how long it takes for the orbit starting at that point to come back in a ball centered about it. Roughly speaking, what is proven in [1] is that these return times scale like the diameter of the ball to a power which is the pointwise dimension on the attractor. A crucial hypothesis to prove their theorem is hyperbolicity of the attractor. In general, recurrence times can only provide a lower bound to pointwise dimension. As pointed out in [5], the use of recurence times as a practical tool to compute the so-called spectrum for generalized dimensions [7] appeared about 20 years ago and was rediscovered (and set explicitely and systematically) in [6]. But the ansatz consisting in replacing the measure of balls by the inverse return time needed to come back into them, though working reasonably well to determine pointwise dimension, can completely fail [2, 6] when used to calculate the spectrum for generalized dimensions. The reason is essentially that return times and measures of balls have different fluctuations at large scales, even in case of very ”nicely” behaved dynamical systems. More recently, hitting times instead of recurrence times have been investigated theoretically [2, 3] and turn out to be more flexible and provide complementary informations. In conclusion, we share the confidence of Halsey and Jensen in the reliability of recurrence times (and hitting time as well) in practical calculations of multifractal characteristics of attractors, but emphasize that this approach, which emerged at a heuristic level, now relies on mathematical results showing the usefulness, but also the limitations, of recurrence and hitting times in capturing multifractality. Needless to say that this field is promising both for experimentalists and mathematicians. 1 CPhT,
CNRS-Ecole polytechnique, France,
[email protected] Matematica Applicata, Universit´ a di Pisa, Italia,
[email protected]
2 Dipartimento
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References [1] Barreira, L. & Saussol, B., Hausdorff dimension of measures via Poincar´e recurrence, Commun. Math. Phys. 219 (2001), 443–463. [2] J.-R. Chazottes, E. Ugalde, Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources, to appear in Discr. & Cont. Dynam. Sys. B. [3] S. Galatolo, Dimension via waiting time and recurrence, preprint downloadable at arXiv: http://lanl.arxiv.org/abs/math.DS/0309223. [4] Gratrix, S. & Elgin, J.N., Pointwise dimensions of the Lorenz attractor, Phys. Rev. Lett. 92, 014101 (2004). [5] Halsey, T.C. & Jensen, M.H., Hurricanes and butterflies, Nature 428, 11 March 2004, 127–128. [6] Haydn, N., Luevano, J., Mantica, G., Vaienti, S., Multifractal properties of return time statistics, Phys. Rev. Lett. 88, No. 22, 3 June 2003. [7] Pesin, Ya., Dimension Theory in Dynamical Systems, University of Chicago Press, Chicago, 1997.
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