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References
Abikoff, W.: The real analytic theory of Teichmüller space, Lect. Notes Math. 820, Springer, 1980.
Abraham, R., Robbin, J.: Transversal Mappings and Flows, Benjamin, 1967.
Adler, R., Konheim, A., McAndrew, M.: Topological entropy, Trans. Amer. Math. Soc. 114, 309–319 (1965).
Ahlfors, L.: Complex Analysis, McGraw-Hill, 1966.
Allwright, D.J.: Hypergraphic functions and bifurcations in recurrence relations, SIAM J. Appl. Math. 34, 687–691 (1978).
Artin, M., Mazur, B.: On periodic points, Annals of Math. 81, 82–99 (1965).
Block, L.: Noncontinuity of topological entropy of maps of the Cantor set and of the interval, Proc. Amer. Math. Soc. 50, 388–393 (1975).
Block, L.: Mappings of the interval with finitely many periodic points have zero entropy, Proc. Amer. Math. Soc. 67, 357–360 (1977).
Block, L.: An example where topological entropy is continuous, Trans. Amer. Math. Soc. 231, 201–213 (1977).
Block, L.: Continuous maps of the interval with finite nonwandering set, Trans. Amer. Math. Soc. 240, 221–230 (1978).
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.
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.
Bowen, R.: Entropy for maps of the interval, Topology 16, 465–467 (1977).
Bowen, R.: On Axiom A Diffeomorphisms, Notes of NSF Regional Conference, North Dakota State Univ., Fargo, N.D., 1977.
Bowen, R.: Bernoulli maps of the interval, Israel J. Math. 28, 161–168 (1977).
Bowen, R., Franks, J.: The periodic points of maps of the disk and the interval, Topology 15, 337–442 (1976).
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.
Collet, P., Eckmann, J.-P.: Iterated Maps of the Interval as Dynamical Systems, Progress in Physics 1, Birkhauser, 1980.
Fatou, M.P.: Sur les équations fonctionnelles, Bull Soc. Math. France 47, 161–271 (1919); 48, 33–94, 208–314 (1920).
Feigenbaum, M.: Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 22–52 (1978); 21, 669–706 (1979).
Feigenbaum, M.: The transition to aperiodic behavior in turbulent systems, Commun. Math. Phys. 77, 65–86 (1980).
Goodman, T.N.T.: Relating topological entropy and measure entropy, Bull. London Math. Soc. 3, 176–180 (1971).
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).
Guckenheimer, J.: On the bifurcation of maps of the interval, Invent. Math. 39, 165–178 (1977).
Guckenheimer, J.: Bifurcations of dynamical systems, C.I.M.E. Lectures, 1978.
Guckenheimer, J.: Sensitive dependence on initial conditions for one-dimensional maps, Comm. Math. Phys. 70, 133–160 (1979).
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.
Jakobson, M.V.: Structure of polynomial mappings on a singular set, Mat. Sbornik 77, 105–124 (1968) (= Math. USSR Sb. 6, 97–114 (1968)).
Jakobson, M.V.: On smooth mappings of the circle into itself, Mat. Sb. 85, 163–188 (1971) (= Math. USSR Sb. 14, 161–185 (1971)).
Jakobson, M.V.: Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Com. Math. Phys. 81, 39–88 (1981).
Jonker, L.: Periodic orbits and kneading invariants, Proc. London Math. Soc. 39, 428–450 (1979).
Jonker, L.: A monotonicity theorem for the family fa(x)=a−x2, Proc. A.M.S. 85, 434–436 (1982).
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).
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).
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).
Julia, G.: Mémoire sur l’iteration des fonctions rationelles, J. de Math. (Liouville), ser. 7, 4, 47–245 (1918).
Kolata, G.B.: Cascading bifurcations: the mathematics of chaos (news article), Science 189, 984–985 (1975).
Lanford, O.E.: Smooth transformations of intervals, Séminaire Bourbaki 563 (1980/81), Lecture Notes Math. 901, Springer 1981.
Lanford, O.E.: A computer-assisted proof of the Feigenbaum conjectures, Bull. A.M.S. 6, 427–434 (1982).
Lang, S.: Differential Manifolds, Addison-Wesley, 1972.
Li, T., Yorke, J.A.: Period three implies chaos, Amer. Math. Monthly 82, 985–992 (1975).
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.
May, R.M.: Biological populations obeying difference equations: stable points, stable cycles and chaos, J. Theor. Biol. 51, 511–524 (1975).
May, R.M.: Simple mathematical models with very complicated dynamics (review article), Nature 261, 459–467 (1976).
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).
Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings, Astérisque 50, 299–310 (1977) and Studia Math. 67, 45–63 (1980).
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.
Parry, W.: Symbolic dynamics and transformations of the unit interval, Trans. Amer. Math. Soc. 122, 368–378 (1966).
Rothschild, J.: On the computation of topological entropy, Thesis, CUNY, 1971.
Ruelle, D.: Applications conservant une mesure absolument continue par rapport a dx sur [0,1], Comm. Math. Phys. 55, 47–51 (1977).
Šarkovskii, A.N.: Coexistence of cycles of a continuous map of a line into itself (Russian), Ukr. mat. Ž. 16, 61–71 (1964).
Singer, D.: Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35, 260–267 (1978).
Smale, S., Williams, R.: The qualitative analysis of a difference equation of population growth, J. Mathematical Biology 3, 1–4 (1976).
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).
Straffin, P.D., Jr.: Periodic points of continuous functions, Math. Mag. 51, 99–105 (1978).
Targonski, G.: Topics in Iteration Theory (Studia Math.: Skript 6), Vandenhoeck u. Ruprecht, Göttingen 1981.
Weil, A.: Numbers of solutions of equations in finite fields, Bull. Amer. Math. Soc. 55, 497–508 (1949).
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.
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Dedicated to the Memory of Rufus Bowen and Peter Štefan.
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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
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