Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T04:06:51.755Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems: 1. The case of negative Schwarzian derivative

Published online by Cambridge University Press:  19 September 2008

M. Yu. Lyubich
M. Yu. Lyubich
Affiliation:
Steklov Mathematical Institute(Leningrad Department), Fontanka 27, Leningrad 191011, USSR
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is proved that an arbitrary one dimensional dynamical system with negative Schwarzian derivative and non-degenerate critical points has no wandering intervals. This result implies a rather complete view of the dynamics of such a system. In particular, every minimal topological attractor is either a limit cycle, or a one dimensional manifold with boundary, or a solenoid. The orbit of a generic point tends to some minimal attractor.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 4 , December 1989 , pp. 737 - 749
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[1]Blokh, A.M.. Decomposition of dynamical systems on an interval. Russian Math. Surv. 38 N5 (1983), 133–134.Google Scholar
[2]Blokh, A.M.. On dynamical systems on one dimensional branched manifolds (in Russian), I, II, III. Theory Fund., Functional Anal & Appl. (Kharkov) 46 (1986), 8–18; 47 (1987), 67–77; 48 (1987), 32–46.Google Scholar
[3]Blokh, A.M. & Lyubich, M. Yu.. Attractors of maps of the interval. Func. Anal. & Appl. 21 N2 (1987), 70–71.Google Scholar
[4]Blokh, A.M. and Lyubich, M. Yu.. On typical behaviour of trajectories of transformations of the interval (in Russian). Theory Fund., Functional Anal. & Appl., Kharkov 49 (1988), 5–16.Google Scholar
[5]Blokh, A.M. & Lyubich, M. Yu.. Ergodicity of transitive unimodal transformations of the interval. Ukranian Math. J., to appear.Google Scholar
[6]Coven, E.M. & Nitecki, Z.. Non-wandering sets of the powers of maps of the interval. Ergot. Th. & Dynam. Sys. 1 Nl (1981), 10–31.Google Scholar
[7]Fatou, P.. Sur les equations fonctionnelles. Bull. Soc. Math. France 48 (1920), 33–94.Google Scholar
[8]Feigenlaum, M.. Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19 (1978), 25–52.Google Scholar
[9]Golusin, G.M.. Geometric Theory of Functions of Complex Variable. GITTL: Moscow-Leningrad, 1952.Google Scholar
[10]Guckenheimer, J.. Sensitive dependence to initial conditions for one dimensional maps. Comm. Math. Phys. 70 N2 (1979), 133–160.CrossRefGoogle Scholar
[11]Guckenheimer, J.. Limit sets of S-unimodal maps with zero entropy Comm. Math. Phys. 110 N4 (1987), 655–659.Google Scholar
[12]Hall, G.R.. A C∞-Denjoy counterexample. Ergod. Th. & Dynam. Sys. 1 N3 (1981), 261–272.Google Scholar
[13]Jacobson, M.V.. On smooth mappings of the circle into itself (in Russian). Math. Sbornik. 14 N2 (1971), 163–188.Google Scholar
[14]Lyubich, M.Ju.. On typical behaviour of the trajectories of a rational mappings of the sphere. Soviet Math. Dokl. 27 (1983), 22–25.Google Scholar
[15]Melo, W.de and Strien, S.van. A structure theorem in one dimensional dynamics. Preprint IMPA, Sér. A-063 (1986).Google Scholar
[16]Milnor, J.. On the concept of attractor. Comm. Math. Phys. 99 (1985), 177–195.Google Scholar
[17]Misiurewicz, M.. Structure of mappings of an interval with zero entropy. Preprint IHES/M/78/249 (1978).Google Scholar
[18]Nitecki, Z.. Differentiate Dynamics. MIT Press: Cambridge, Massachusetts-London, 1971.Google Scholar
[19]Sharkovskii, A.N.. Partially ordered system of limit sets (in Russian). Soviet Math. Dokl. 170 (1966), 1276–1278.Google Scholar
[20]Singer, D.. Stable orbits and bifurcations of maps of the interval. SI AM J. Appl. Math. 35 (1978), 260–267.Google Scholar
[21]Sullivan, D.. Quasi conformal homeomorphisms and dynamics. I. Ann. of Math. 122 N3 (1985), 401–418.Google Scholar
[22]Strien, S.van. On the bifurcations creating horseshoes. Springer Led. Notes Math. 898 (1981), 316–351.CrossRefGoogle Scholar
[23]Schwartz, A.. A generalization of Poincare-Bendixon theorem on closed two-dimensional manifolds. Amer. J. Math. 85 (1963), 453–458.Google Scholar
[24]Yoccoz, J.C.. Il ny pas de coutre-exemple de Denjoy analitique. C.R. Acad. Sci. Paris. Ser. 1 N7 298 (1984), 141–144.Google Scholar
[25]Young, L.S.. A closing lemma on the interval. Inv. Math. 54 N2 (1979), 179–187.Google Scholar
[26]Jonker, L. and Rand, D. A.. Bifurcations in one dimension, I: The non-wandering set. Inventions Math. 62 (1981), 347–365.CrossRefGoogle Scholar