Abstract
We consider itcrates of maps of an interval to itself and their stable periodic orbits. When these maps depend on a parameter, one can observe period doubling bifurcations as the parameter is varied. We investigate rigorously those aspects of these bifurcations which are universal, i.e. independent of the choice of a particular one-parameter family. We point out that this universality extends to many other situations such as certain chaotic regimes. We describe the ergodic properties of the maps for which the parameter value equals the limit of the bifurcation points.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Collet, P., Eckmann, J.-P.: A renormalization group analysis of the hierarchical model in statistical physics. Lecture Notes in Physics, Vol. 74. Berlin, Heidelberg, New York: Springer 1978
Dieudonné, J.: Foundations of modern analysis. New York, London: Academic Press 1969
Feigenbaum, M.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys.19, 25 (1978);21, 669 (1979)
Guckenheimer, J.: Bifurcations of dynamical systems. C.I.M.E. Lectures 1978
Hartmann, P.: Ordinary differential equations. New York, London: Wiley 1964
Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Berlin, Heidelberg, New York: Springer 1977
Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966
Lanford III, O.E.: To appear, and Lecture Notes in Physics, Vol. 116, Berlin, Heidelberg, New York: Springer 1980
May, R.M.: Simple mathematical models with very complicated dynamics. Nature261, 259–467 (1976)
Misiurewicz, M.: Structure of mappings of the interval with zero entropy. Preprint I.H.E.S. (1978)
Misiurewicz, M.: Absolutely continuous measures for certain maps on an interval. Preprint I.H.E.S./M/79/293 (1979)
Singer, D.: Stable orbits and bifurcations of maps of the interval. S.I.A.M. J. Appl. Math.35, 260–267 (1978)
Stefan, P.: A theorem of Sharkovskii on the existence of periodic orbits of continuous endomorphisms of the real line. Commun. Math. Phys.54, 237–248 (1977)
Collet, P., Eckmann, J.-P.: Iterated maps on the interval as dynamical systems. Progress in Physics. Birkh äuser Boston 1980 (to appear)
Author information
Authors and Affiliations
Additional information
Communicated by D. Ruelle
Rights and permissions
About this article
Cite this article
Collet, P., Eckmann, J.P. & Lanford, O.E. Universal properties of maps on an interval. Commun.Math. Phys. 76, 211–254 (1980). https://doi.org/10.1007/BF02193555
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02193555