Skip to main content
SpringerLink
Log in

On iterated maps of the interval

  • Conference paper
  • First Online:
Book cover Dynamical Systems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1342))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abikoff, W.: The real analytic theory of Teichmüller space, Lect. Notes Math. 820, Springer, 1980.

    Google Scholar 

  • Abraham, R., Robbin, J.: Transversal Mappings and Flows, Benjamin, 1967.

    Google Scholar 

  • Adler, R., Konheim, A., McAndrew, M.: Topological entropy, Trans. Amer. Math. Soc. 114, 309–319 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  • Ahlfors, L.: Complex Analysis, McGraw-Hill, 1966.

    Google Scholar 

  • Allwright, D.J.: Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math. 34, 687–691 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  • Artin, M., Mazur, B.: On periodic points, Annals of Math. 81, 82–99 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  • Block, L.: Noncontinuity of topological entropy of maps of the Cantor set and of the interval, Proc. Amer. Math. Soc. 50, 388–393 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • Block, L.: Mappings of the interval with finitely many periodic points have zero entropy, Proc. Amer. Math. Soc. 67, 357–360 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • Block, L.: An example where topological entropy is continuous, Trans. Amer. Math. Soc. 231, 201–213 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • Block, L.: Continuous maps of the interval with finite nonwandering set, Trans. Amer. Math. Soc. 240, 221–230 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

  • Bowen, R.: Entropy for maps of the interval, Topology 16, 465–467 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • Bowen, R.: On Axiom A Diffeomorphisms, Notes of NSF Regional Conference, North Dakota State Univ., Fargo, N.D., 1977.

    Google Scholar 

  • Bowen, R.: Bernoulli maps of the interval, Israel J. Math. 28, 161–168 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • Bowen, R., Franks, J.: The periodic points of maps of the disk and the interval, Topology 15, 337–442 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Collet, P., Eckmann, J.-P.: Iterated Maps of the Interval as Dynamical Systems, Progress in Physics 1, Birkhauser, 1980.

    Google Scholar 

  • Fatou, M.P.: Sur les équations fonctionnelles, Bull Soc. Math. France 47, 161–271 (1919); 48, 33–94, 208–314 (1920).

    MathSciNet  MATH  Google Scholar 

  • Feigenbaum, M.: Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 22–52 (1978); 21, 669–706 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  • Feigenbaum, M.: The transition to aperiodic behavior in turbulent systems, Commun. Math. Phys. 77, 65–86 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  • Goodman, T.N.T.: Relating topological entropy and measure entropy, Bull. London Math. Soc. 3, 176–180 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Guckenheimer, J.: On the bifurcation of maps of the interval, Invent. Math. 39, 165–178 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • Guckenheimer, J.: Bifurcations of dynamical systems, C.I.M.E. Lectures, 1978.

    Google Scholar 

  • Guckenheimer, J.: Sensitive dependence on initial conditions for one-dimensional maps, Comm. Math. Phys. 70, 133–160 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Jakobson, M.V.: Structure of polynomial mappings on a singular set, Mat. Sbornik 77, 105–124 (1968) (= Math. USSR Sb. 6, 97–114 (1968)).

    MathSciNet  Google Scholar 

  • Jakobson, M.V.: On smooth mappings of the circle into itself, Mat. Sb. 85, 163–188 (1971) (= Math. USSR Sb. 14, 161–185 (1971)).

    MathSciNet  Google Scholar 

  • Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Com. Math. Phys. 81, 39–88 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  • Jonker, L.: Periodic orbits and kneading invariants, Proc. London Math. Soc. 39, 428–450 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  • Jonker, L.: A monotonicity theorem for the family fa(x)=a−x2, Proc. A.M.S. 85, 434–436 (1982).

    MathSciNet  MATH  Google Scholar 

  • 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 

  • 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).

    Article  MathSciNet  MATH  Google Scholar 

  • 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).

    Article  MathSciNet  MATH  Google Scholar 

  • Julia, G.: Mémoire sur l’iteration des fonctions rationelles, J. de Math. (Liouville), ser. 7, 4, 47–245 (1918).

    MATH  Google Scholar 

  • Kolata, G.B.: Cascading bifurcations: the mathematics of chaos (news article), Science 189, 984–985 (1975).

    Article  Google Scholar 

  • Lanford, O.E.: Smooth transformations of intervals, Séminaire Bourbaki 563 (1980/81), Lecture Notes Math. 901, Springer 1981.

    Google Scholar 

  • Lanford, O.E.: A computer-assisted proof of the Feigenbaum conjectures, Bull. A.M.S. 6, 427–434 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  • Lang, S.: Differential Manifolds, Addison-Wesley, 1972.

    Google Scholar 

  • Li, T., Yorke, J.A.: Period three implies chaos, Amer. Math. Monthly 82, 985–992 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • May, R.M.: Biological populations obeying difference equations: stable points, stable cycles and chaos, J. Theor. Biol. 51, 511–524 (1975).

    Article  Google Scholar 

  • May, R.M.: Simple mathematical models with very complicated dynamics (review article), Nature 261, 459–467 (1976).

    Article  Google Scholar 

  • 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).

    Article  MathSciNet  MATH  Google Scholar 

  • Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings, Astérisque 50, 299–310 (1977) and Studia Math. 67, 45–63 (1980).

    MathSciNet  MATH  Google Scholar 

  • 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 

  • Parry, W.: Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122, 368–378 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  • Rothschild, J.: On the computation of topological entropy, Thesis, CUNY, 1971.

    Google Scholar 

  • Ruelle, D.: Applications conservant une mesure absolument continue par rapport a dx sur [0,1], Comm. Math. Phys. 55, 47–51 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Singer, D.: Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35, 260–267 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  • Smale, S., Williams, R.: The qualitative analysis of a difference equation of population growth, J. Mathematical Biology 3, 1–4 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  • 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).

    Article  MATH  Google Scholar 

  • Straffin, P.D., Jr.: Periodic points of continuous functions, Math. Mag. 51, 99–105 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  • Targonski, G.: Topics in Iteration Theory (Studia Math.: Skript 6), Vandenhoeck u. Ruprecht, Göttingen 1981.

    Google Scholar 

  • Weil, A.: Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55, 497–508 (1949).

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

Download references

Author information

Authors and Affiliations

  1. Institute for Advanced Study, 08540, Princeton, NJ, USA

    John Milnor

  2. Princeton University, 08544, Princeton, NJ, USA

    William Thurston

Authors
  1. John Milnor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William Thurston
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

James C. Alexander

Additional information

Dedicated to the Memory of Rufus Bowen and Peter Štefan.

Rights and permissions

Reprints and permissions

Copyright information

© 1988 Springer-Verlag

About this paper

Cite this paper

Milnor, J., Thurston, W. (1988). On iterated maps of the interval. In: Alexander, J.C. (eds) Dynamical Systems. Lecture Notes in Mathematics, vol 1342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082847

Download citation

  • DOI: https://doi.org/10.1007/BFb0082847

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50174-9

  • Online ISBN: 978-3-540-45946-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation