On iterated maps of the interval

  • John Milnor
  • William Thurston
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1342)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abikoff, W.: The real analytic theory of Teichmüller space, Lect. Notes Math. 820, Springer, 1980.Google Scholar
  2. Abraham, R., Robbin, J.: Transversal Mappings and Flows, Benjamin, 1967.Google Scholar
  3. Adler, R., Konheim, A., McAndrew, M.: Topological entropy, Trans. Amer. Math. Soc. 114, 309–319 (1965).MathSciNetCrossRefMATHGoogle Scholar
  4. Ahlfors, L.: Complex Analysis, McGraw-Hill, 1966.Google Scholar
  5. Allwright, D.J.: Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math. 34, 687–691 (1978).MathSciNetCrossRefMATHGoogle Scholar
  6. Artin, M., Mazur, B.: On periodic points, Annals of Math. 81, 82–99 (1965).MathSciNetCrossRefMATHGoogle Scholar
  7. Block, L.: Noncontinuity of topological entropy of maps of the Cantor set and of the interval, Proc. Amer. Math. Soc. 50, 388–393 (1975).MathSciNetCrossRefMATHGoogle Scholar
  8. Block, L.: Mappings of the interval with finitely many periodic points have zero entropy, Proc. Amer. Math. Soc. 67, 357–360 (1977).MathSciNetCrossRefMATHGoogle Scholar
  9. Block, L.: An example where topological entropy is continuous, Trans. Amer. Math. Soc. 231, 201–213 (1977).MathSciNetCrossRefMATHGoogle Scholar
  10. Block, L.: Continuous maps of the interval with finite nonwandering set, Trans. Amer. Math. Soc. 240, 221–230 (1978).MathSciNetCrossRefMATHGoogle Scholar
  11. Block, L., Guckenheimer, J., Misiurewicz, M., Young, L.-S.: Periodic points and topological entropy of one dimensional maps, pp. 18–34 of Global Theory of Dynamical Systems, ed. Nitecki and Robinson, Lecture Notes in Math. 819, Springer 1980.Google Scholar
  12. Bowen, R.: Topological entropy and Axiom A, pp. 23–41 of Global Analysis, ed. Chern and Smale, Proc. Symp. Pure Math. 14, A.M.S., 1970.Google Scholar
  13. Bowen, R.: Entropy for maps of the interval, Topology 16, 465–467 (1977).MathSciNetCrossRefMATHGoogle Scholar
  14. Bowen, R.: On Axiom A Diffeomorphisms, Notes of NSF Regional Conference, North Dakota State Univ., Fargo, N.D., 1977.Google Scholar
  15. Bowen, R.: Bernoulli maps of the interval, Israel J. Math. 28, 161–168 (1977).MathSciNetCrossRefMATHGoogle Scholar
  16. Bowen, R., Franks, J.: The periodic points of maps of the disk and the interval, Topology 15, 337–442 (1976).MathSciNetCrossRefMATHGoogle Scholar
  17. Bowen, R., Lanford, O.E., III: Zeta functions of restrictions of the shift transformation, pp. 43–49 of Global Analysis, ed. Chern and Smale, Proc. Symp. Pure Math. 14, A.M.S., 1970.Google Scholar
  18. Collet, P., Eckmann, J.-P.: Iterated Maps of the Interval as Dynamical Systems, Progress in Physics 1, Birkhauser, 1980.Google Scholar
  19. Fatou, M.P.: Sur les équations fonctionnelles, Bull Soc. Math. France 47, 161–271 (1919); 48, 33–94, 208–314 (1920).MathSciNetMATHGoogle Scholar
  20. Feigenbaum, M.: Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 22–52 (1978); 21, 669–706 (1979).MathSciNetCrossRefMATHGoogle Scholar
  21. Feigenbaum, M.: The transition to aperiodic behavior in turbulent systems, Commun. Math. Phys. 77, 65–86 (1980).MathSciNetCrossRefMATHGoogle Scholar
  22. Goodman, T.N.T.: Relating topological entropy and measure entropy, Bull. London Math. Soc. 3, 176–180 (1971).MathSciNetCrossRefMATHGoogle Scholar
  23. Guckenheimer, J.: Endomorphisms of the Riemann sphere, pp. 95–123 of Global Analysis, ed. Chern and Smale, Proc. Symp. Pure Math. 14, A.M.S. (1970).Google Scholar
  24. Guckenheimer, J.: On the bifurcation of maps of the interval, Invent. Math. 39, 165–178 (1977).MathSciNetCrossRefMATHGoogle Scholar
  25. Guckenheimer, J.: Bifurcations of dynamical systems, C.I.M.E. Lectures, 1978.Google Scholar
  26. Guckenheimer, J.: Sensitive dependence on initial conditions for one-dimensional maps, Comm. Math. Phys. 70, 133–160 (1979).MathSciNetCrossRefMATHGoogle Scholar
  27. Guckenheimer, J.: The growth of topological entropy for one-dimensional maps, pp. 216–223 of Global Theory of Dynamical Systems, ed. Nitecki and Robinson, Lecture Notes in Math. 819, Springer 1980.Google Scholar
  28. Jakobson, M.V.: Structure of polynomial mappings on a singular set, Mat. Sbornik 77, 105–124 (1968) (= Math. USSR Sb. 6, 97–114 (1968)).MathSciNetGoogle Scholar
  29. Jakobson, M.V.: On smooth mappings of the circle into itself, Mat. Sb. 85, 163–188 (1971) (= Math. USSR Sb. 14, 161–185 (1971)).MathSciNetGoogle Scholar
  30. Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Com. Math. Phys. 81, 39–88 (1981).MathSciNetCrossRefMATHGoogle Scholar
  31. Jonker, L.: Periodic orbits and kneading invariants, Proc. London Math. Soc. 39, 428–450 (1979).MathSciNetCrossRefMATHGoogle Scholar
  32. Jonker, L.: A monotonicity theorem for the family fa(x)=a−x2, Proc. A.M.S. 85, 434–436 (1982).MathSciNetMATHGoogle Scholar
  33. Jonker, L., Rand, D.: A lower bound for the entropy of certain maps of the unit interval, preprint, Kingston, Ontario, and Univ. of Warwick (1978).Google Scholar
  34. Jonker, L., Rand, D.: The periodic orbits and entropy of certain maps of the unit interval, J. London Math. Soc. (2), 22, 175–181 (1980).MathSciNetCrossRefMATHGoogle Scholar
  35. Jonker, L., Rand, D.: Bifurcations in one dimension; I, The nonwandering set, Invent. Math. 62, 347–365 (1981); II, A versal model for bifurcations, Invent. Math. 63, 1–15 (1981).MathSciNetCrossRefMATHGoogle Scholar
  36. Julia, G.: Mémoire sur l’iteration des fonctions rationelles, J. de Math. (Liouville), ser. 7, 4, 47–245 (1918).MATHGoogle Scholar
  37. Kolata, G.B.: Cascading bifurcations: the mathematics of chaos (news article), Science 189, 984–985 (1975).CrossRefGoogle Scholar
  38. Lanford, O.E.: Smooth transformations of intervals, Séminaire Bourbaki 563 (1980/81), Lecture Notes Math. 901, Springer 1981.Google Scholar
  39. Lanford, O.E.: A computer-assisted proof of the Feigenbaum conjectures, Bull. A.M.S. 6, 427–434 (1982).MathSciNetCrossRefMATHGoogle Scholar
  40. Lang, S.: Differential Manifolds, Addison-Wesley, 1972.Google Scholar
  41. Li, T., Yorke, J.A.: Period three implies chaos, Amer. Math. Monthly 82, 985–992 (1975).MathSciNetCrossRefMATHGoogle Scholar
  42. Lorenz, E.: On the prevalence of aperiodicity in simple systems, pp. 53–57 of Global Analysis, ed. Grmela and Marsden, Lecture Notes in Math. 755, Springer 1979.Google Scholar
  43. May, R.M.: Biological populations obeying difference equations: stable points, stable cycles and chaos, J. Theor. Biol. 51, 511–524 (1975).CrossRefGoogle Scholar
  44. May, R.M.: Simple mathematical models with very complicated dynamics (review article), Nature 261, 459–467 (1976).CrossRefGoogle Scholar
  45. Metropolis, N., Stein, M.L., Stein, P.R.: On finite limit sets for transformations on the unit interval, Journal of Combinatorial Theory (A) 15, 25–44 (1973).MathSciNetCrossRefMATHGoogle Scholar
  46. Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings, Astérisque 50, 299–310 (1977) and Studia Math. 67, 45–63 (1980).MathSciNetMATHGoogle Scholar
  47. Oster, G., Guckenheimer, J.: Bifurcation phenomena in population models, pp. 327–353 of "The Hopf Bifurcation and its Applications," by Marsden and McCracken, Springer 1976.Google Scholar
  48. Parry, W.: Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122, 368–378 (1966).MathSciNetCrossRefMATHGoogle Scholar
  49. Rothschild, J.: On the computation of topological entropy, Thesis, CUNY, 1971.Google Scholar
  50. Ruelle, D.: Applications conservant une mesure absolument continue par rapport a dx sur [0,1], Comm. Math. Phys. 55, 47–51 (1977).MathSciNetCrossRefMATHGoogle Scholar
  51. Šarkovskii, A.N.: Coexistence of cycles of a continuous map of a line into itself (Russian), Ukr. mat. Ž. 16, 61–71 (1964).Google Scholar
  52. Singer, D.: Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35, 260–267 (1978).MathSciNetCrossRefMATHGoogle Scholar
  53. Smale, S., Williams, R.: The qualitative analysis of a difference equation of population growth, J. Mathematical Biology 3, 1–4 (1976).MathSciNetCrossRefMATHGoogle Scholar
  54. Štefan, P.: A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line, Com. Math. Phys. 54, 237–248 (1977).CrossRefMATHGoogle Scholar
  55. Straffin, P.D., Jr.: Periodic points of continuous functions, Math. Mag. 51, 99–105 (1978).MathSciNetCrossRefMATHGoogle Scholar
  56. Targonski, G.: Topics in Iteration Theory (Studia Math.: Skript 6), Vandenhoeck u. Ruprecht, Göttingen 1981.Google Scholar
  57. Weil, A.: Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55, 497–508 (1949).MathSciNetCrossRefMATHGoogle Scholar
  58. Williams, R.F.: The zeta function in global analysis, pp. 335–339 of Global Analysis, ed. Chern and Smale, Proc. Symp. Pure Math. 14, A.M.S. 1970.Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • John Milnor
    • 1
  • William Thurston
    • 2
  1. 1.Institute for Advanced StudyPrincetonUSA
  2. 2.Princeton UniversityPrincetonUSA

Personalised recommendations