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Multifractal Spectrum of an Experimental (Video Feedback) Farey Tree

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Abstract

A camera films a screen to which it is connected. It films its own image, feeding back the image to the screen. The camera can turn around an optical axis. A pattern of p light spots on the screen and q turns (feedback loops) of the camera appears, where p and q follow the hierarchy of a Farey tree. The Farey tree induces a measure distribution μ on the unit segment, different from the hyperbolic one μ H induced by the Farey-Brocot interpolation. In this paper the multifractal spectrum of μ is studied and compared with that of μ H; the study of the latter spectrum is refined. The spectra are studied in this paper by means of different tools from Number Theory. The results of this study are interpreted in terms of p and q, empirically obtained in the video feedback experiment.

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References

  1. B. Essevez Roulet, P. Petitjeans, J. E. Wesfreid and M. Rosen, Farey sequences of spatiotemporal patterns in video feedback. Phys. Rev. E 61(4): 3743–49 (2000) (http://www.videofeedback.dk).

    Google Scholar 

  2. M. M. Dodson, Exceptional. sets in dynamical systems and Diophantine approximation, arXiv:math:NT/0108210 V1, Los Alamos National Laboratory, xxx.lanl.gov (2001)

  3. P. Berge, Y. Pomeau and C. Vidal, Lordre dans le chaos, Collection Enseignements des Sciences, (Ed. Hermann, Paris, 1994).

  4. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia and B. Schraiman, Fractal measures and their singularities: The characterization of strange sets, Nucl. Phys. B (Proc. Suppl.) 2: 513–516 (1986).

    Google Scholar 

  5. M. Duong-Van, Phase transition of multifractals. Nucl. Phys. B (Proc. Suppl.) 2: 521–526 (1987).

    Google Scholar 

  6. R. Cawley and R. D. Mauldin, Multifractal decompositions of Moran fractals. Adv. Math. 92(2): 196–236 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Riedi and B. Mandelbrot, The inversion formula for continuous multifractals. Adv. Appl. Math. 19: 332–354 (1997).

    Google Scholar 

  8. R. Riedi and B. Mandelbrot, Exception to the multifractal formalism for discontinuous measures. Math. Proc. Camb. Phil. Soc. 123: 133–157 (1998).

    Google Scholar 

  9. M. Piacquadio and E. Cesaratto, Multifractal spectrum and thermodynamical formalism of the Farey tree. Int. J. Bifurcation and Chaos 11(5): 1331–1358 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  10. G. H. Hardy and E. M. Wright. An introduction to the Theory of Numbers, Chap. I-IV, (Clarendon Press, Oxford, 1938).

  11. S. Grynberg and M. Piacquadio, Hyperbolic geometry and multifractal spectra. Part. II. Trabajos de Matemáticas 252, Publicaciones Previas del Instituto Argentino de Matemáticas, I.A.M.-CONICET (1995).

  12. P. Cvitanovic, H. Jensen, L. P. Kadanoff and I. Procaccia, Renormalization, unstable manifolds, and the fractal structure of mode locking. Phys. Rev. Lett. 55(4): 343–346 (1985).

    Google Scholar 

  13. C. Series, Non Euclidean geometry, continued fractions and ergodic theory. Math. Intell. 4(1): 24–28 (1982).

  14. M. Piacquadio Losada and S. Grynberg, Cantor staircases in physics and Diophantine approximation. Int. J. Bifurcation and Chaos 8(6): 1095–1106 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Grynberg and M. Piacquadio, Self-similarity of Farey staircases. arXiv:math-ph/0306024 V1, Los Alamos National Laboratory, xxx.lanl.gov. (2003).

  16. V. Jarník, Zur metrischen Theorie der Diophantischen Approximationen. Prace Mat-Fiz 91–106 (1928–29).

  17. C. Series, The Markov spectrum in the Hecke group G5. Proc. London Math. Soc. 57: 151–180 (1988).

    Google Scholar 

  18. A. Haas and C. Series, The Hurwitz constant and Diophantine approximation on Hecke groups. J. London Math. Soc. 34: 219–234 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  19. C. Series, The modular surface and continued fractions. J. London Math. Soc. (2) 31(1): 69–80 (1985).

    Google Scholar 

  20. M. Piacquadio, The geometry of Farey staircases. Int. J. Bifurcation and Chaos 14(12): 4075–4096 (2004).

    Article  MATH  Google Scholar 

  21. E. Cesaratto and M. Piacquadio, Multifractal formalism of the Farey partition. Revista de la Unión Matemática Argentina 41(2): 51–66 (1998).

    MATH  Google Scholar 

  22. L. J. Good, The fractional dimensional theory of continued fractions. Proc. Cam. Phil. Soc. 37: 199–228 (1941).

    Article  MATH  MathSciNet  Google Scholar 

  23. G. H. Hardy and E. M. Wright. An introduction to the Theory of Numbers, Chap. XVIII (Clarendon Press, Oxford, 1938).

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Authors and Affiliations

  1. Grupo de Medios Porosos, Departamento de Física, Facultad de Ingeniería, Universidad de Buenos Aires, Paseo Colón 850 (1063), Buenos Aires, Argentina

    M. Piacquadio & M. Rosen

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  1. M. Piacquadio
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  2. M. Rosen
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Correspondence to M. Piacquadio.

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Piacquadio, M., Rosen, M. Multifractal Spectrum of an Experimental (Video Feedback) Farey Tree. J Stat Phys 127, 783–804 (2007). https://doi.org/10.1007/s10955-006-9217-5

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  • DOI: https://doi.org/10.1007/s10955-006-9217-5

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