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Transactions of the American Mathematical Society

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Genealogy of periodic points of maps of the interval

Author: Robert L. Devaney
Journal: Trans. Amer. Math. Soc. 265 (1981), 137-146
MSC: Primary 58F20; Secondary 28D99, 58F14
MathSciNet review: 607112
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Abstract: We describe the behavior of families of periodic points in one parameter families of maps of the interval which feature a transition from simple dynamics with finitely many periodic points to chaotic mappings. In particular, we give topological criteria for the appearance and disappearance of these families. Our results apply specifically to quadratic maps of the form $ {F_\mu }(x) = \mu x(1 - x)$.

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  • [1] R. Devaney and Z. Nitecki, Shift automorphisms in the Hénon mapping, Comm. Math. Phys. 67 (1979), no. 2, 137–146. MR 539548
  • [2] John Guckenheimer, On the bifurcation of maps of the interval, Invent. Math. 39 (1977), no. 2, 165–178. MR 0438399
  • [3] -, The bifurcation of quadradic functions, Bifurcation Theory and Applications in Scientific Disciplines, Ann. N. Y. Acad. Sci. 316 (1979), 78-85.
  • [4] John Guckenheimer, Bifurcations of dynamical systems, Dynamical systems (C.I.M.E. Summer School, Bressanone, 1978) Progr. Math., vol. 8, Birkhäuser, Boston, Mass., 1980, pp. 115–231. MR 589591
  • [5] Leo Jonker, Periodic orbits and kneading invariants, Proc. London Math. Soc. (3) 39 (1979), no. 3, 428–450. MR 550078, 10.1112/plms/s3-39.3.428
  • [6] John Milnor and William Thurston, On iterated maps of the interval, Dynamical systems (College Park, MD, 1986–87) Lecture Notes in Math., vol. 1342, Springer, Berlin, 1988, pp. 465–563. MR 970571, 10.1007/BFb0082847
  • [7] Jürgen Moser, Stable and random motions in dynamical systems, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J; Annals of Mathematics Studies, No. 77. MR 0442980
  • [8] Zbigniew Nitecki, Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms, The M.I.T. Press, Cambridge, Mass.-London, 1971. MR 0649788
  • [9] Stephen Smale, Diffeomorphisms with many periodic points, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 63–80. MR 0182020

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Article copyright: © Copyright 1981 American Mathematical Society