Abstract
This paper studies the iteration of maps of the interval which have negative Schwarzian derivative and one critical point. The maps in this class are classified up to topological equivalence. The equivalence classes of maps which display sensitivity to initial conditions for large sets of initial conditions are characterized.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allwright, D.: Hypergraphic functions and bifurcation in recurrence relations. SIAM J. Appl. Math.34, 687–691 (1978)
Block, L., Franke, J.: Existence of periodic points for maps ofS 1. Inventiones Math.22, 69–73 (1973)
Bowen, R.: Invariant measures for Markov maps of the interval. Mimeographed, Berkeley 1978
Cartan, E.: Leçons sur la théorie des espaces à connexions projective. Cahiers Scientifiques, Fascicule XVII. Paris: Gauthier-Villar 1937
Denjoy, A.: Sur les courbes définies par les équations différentielles à la surface du tore. J. Math. Pures Appl.9, 333–375 (1932)
Guckenheimer, J.: On the bifurcation of maps of the interval. Inventiones Math.39, 165–178 (1977)
Guckenheimer, J.: Bifurcations of dynamical systems. C.I.M.E. Lectures, 1978
Guckenheimer, J., Oster, G., Ipaktchi, A.: Dynamics of density dependent population models. J. Math. Biol.4, 101–147 (1977)
Hénon, M.: A two dimensional mapping with a strange attractor. Commun. Math. Phys.50, 69–77 (1976)
Herman, M.: Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Thesis, Orsay 1976
Holmes, P., Moon, F.: A magnetoelastic strange attractor. Preprint, Cornell 1979
Jakobson, M.: On smooth mappings of the circle into itself. Math. USSR sb.14, 161–185 (1971)
Jakobson, M.: Topological and metric properties of one dimensional endomorphisms. Doklady Akademia Nauk USSR243, 866–869 (1978)
Jonker, L., Rand, D.: A lower bound for the entropy of certain maps of the unit interval. Preprint, Warwick 1978
Jonker, L., Rand, D.: Bifurcations in one dimension. I, II. Preprint, Warwick 1979
Lorenz, E.: Deterministic non periodic flow. J. Atmos. Sci20, 130–141 (1963)
Milnor, J., Thurston, W.: On iterated maps of the interval. I, II. Preprint, Princeton 1977
Misiurewicz, M.: Structure of mappings of the interval with zero entropy. Preprint, I.H.E.S. 1978
Misiurewicz, M., Szlenk, W.: Entropy of piecewise monotone mappings. Asterisque50, 299–310 (1977)
Parry, W.: Symbolic dynamics and transformations of the unit interval. Trans. Am. Math. Soc.122, 368–378 (1966)
Ruelle, D.: Sensitive dependence on initial conditions and turbulent behaviour of dynamical systems. Ann. N.Y. Acad. Sci. (to appear)
Ruelle, D.: On measures which describe turbulence. Preprint, I.H.E.S. 1979
Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys.20, 167–192 (1971);23, 343 (1971)
Shaw, R.: Strange attractors, chaotic behavior, and information flow. Preprint, Santa-Cruz (1978)
Singer, D.: Stable orbits and bifurcations of maps of the interval. SIAM J. Appl. Math.35, 260–267 (1978)
Spiegel, E., Marzec, C.: A strange attractor. Preprint, 1978
Author information
Authors and Affiliations
Additional information
Communicated by D. Ruelle
Research partially supported by the National Science Foundation and the Volkswagen Foundation. We gratefully acknowledge this support and the hospitality of the Courant Institute for Mathematical Sciences and the Institut des Hautes Études Scientifiques
Rights and permissions
About this article
Cite this article
Guckenheimer, J. Sensitive dependence to initial conditions for one dimensional maps. Commun.Math. Phys. 70, 133–160 (1979). https://doi.org/10.1007/BF01982351
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01982351