Abstract
According to the theory of Schröder and Siegel, certain complex analytic maps possess a family of closed invariant curves in the complex plane. We have made a numerical study of these curves by iterating the map, and have found that the largest curve is a fractal. When the winding number of the map is the golden mean, the fractal curve has universal scaling properties, and the scaling parameter differs from those found for other types of maps. Also, for this winding number, there are universal scaling functions which describe the behaviour asn→∞ of theQ th n iterates of the map, whereQ n is then th Fibonacci number.
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References
Eckmann, J.-P.: Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys.53, 643–654 (1981); Ott, E.: Strange attractors and chaotic motions of dynamical systems. Rev. Mod. Phys.53, 655–671 (1981); Helleman, R.H.G.: In: Fundamental problems in statistical mechanics, Cohen, E.G.D. (ed.). Amsterdam: North Holland 1980; These are three recent review articles which include references to earlier work
Shenker, S.J.: Scaling behaviour in a map of a circle into itself: empirical results. Physica5 D, 405–411 (1982); Ostlund, S., Rand, D., Sethna, J., Siggia, E.: A universal transition from quasiperiodicity to chaos in dissipative systems. Phys. Rev. Lett.49, 132–135 (1982)
Kadanoff, L.P.: Scaling for a critical Kolmogorov-Arnold-Moser trajectory. Phys. Rev. Lett.47, 1641–1643 (1981); Shenker, S.J., Kadanoff, L.P.: Critical behaviour of a KAM surface: 1. empirical results. J. Stat. Phys.27, 631–656 (1982); Widom, M., Kadanoff, L.P.: Renormalization group analysis of bifurcations in area-preserving maps. Physica5 D, 287–292 (1982)
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys.20, 167–192 (1971); Newhouse, S., Ruelle, D., Takens, F.: Occurence of strange axiom A attractors near quasi periodic flows onT m,m≧3. Commun. Math. Phys.64, 35–40 (1978)
Schröder, E.: Über iterirte Funktionen. Math. Ann.3, 296–322 (1871); Siegel, C.L.: Iteration of analytic functions. Ann. Math.43, 607–612 (1942); For a modern treatment, see Moser, J.K., Siegel, C.L.: Lectures on celestial mechanics. New York: Springer 1971
Fatou, P.: Sur les équations fonctionelles. Bull. Soc. Math. France47, 161–271 (1919),48, 33–94 and 208–314 (1920)
Julia, G.: Mémoire sur l'itération des fonctions rationelles. J. Math. Pures Appl.4, 47–245 (1918)
Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat. Astron. Pys.6, 103–144 (1965)
Mandelbrot, B.: Fractal aspects of the iteration ofz → λz(1 −z) for complex λ andz. Ann. N. Y. Acad. Sci.357, 249–259 (1980)
Ruelle, D.: Repellers for real analytic maps. IHES preprint 81/52
Douady, A., Hubbard, J.M.: Itération des polynômes quadratiques complexes. C. R. Acad. Sci. (Paris) Sér. 1,294, 123–126 (1982)
Moser, J.: On invariant curves of area-preserving maps of an annulus. Nachr. Akad. Wiss., Göttingen, Math. Phys. Kl. 1–20 (1962)
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Communicated by O. E. Lanford
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Manton, N.S., Nauenberg, M. Universal scaling behaviour for iterated maps in the complex plane. Commun.Math. Phys. 89, 555–570 (1983). https://doi.org/10.1007/BF01214743
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DOI: https://doi.org/10.1007/BF01214743