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    Okniński, A. Rynio, R. and Peinke, J. 1995. Symmetry-breaking and fractal dependence on initial conditions in dynamical systems: One-dimensional noninvertible mappings. Chaos, Solitons & Fractals, Vol. 5, Issue. 5, p. 783.


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  • Ergodic Theory and Dynamical Systems, Volume 9, Issue 4
  • December 1989, pp. 737-749

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

  • M. Yu. Lyubich (a1)
  • DOI: http://dx.doi.org/10.1017/S0143385700005307
  • Published online: 01 September 2008
Abstract
Abstract

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.

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[1]A.M. Blokh . Decomposition of dynamical systems on an interval. Russian Math. Surv. 38 N5 (1983), 133134.

[3]A.M. Blokh & M. Yu. Lyubich . Attractors of maps of the interval. Func. Anal. & Appl. 21 N2 (1987), 7071.

[8]M. Feigenlaum . Quantitative universality for a class of non-linear transformations. J. Stat. Phys. 19 (1978), 2552.

[10]J. Guckenheimer . Sensitive dependence to initial conditions for one dimensional maps. Comm. Math. Phys. 70 N2 (1979), 133160.

[11]J. Guckenheimer . Limit sets of S-unimodal maps with zero entropy Comm. Math. Phys. 110 N4 (1987), 655659.

[16]J. Milnor . On the concept of attractor. Comm. Math. Phys. 99 (1985), 177195.

[20]D. Singer . Stable orbits and bifurcations of maps of the interval. SI AM J. Appl. Math. 35 (1978), 260267.

[21]D. Sullivan . Quasi conformal homeomorphisms and dynamics. I. Ann. of Math. 122 N3 (1985), 401418.

[22]S.van Strien . On the bifurcations creating horseshoes. Springer Led. Notes Math. 898 (1981), 316351.

[23]A. Schwartz . A generalization of Poincare-Bendixon theorem on closed two-dimensional manifolds. Amer. J. Math. 85 (1963), 453458.

[25]L.S. Young . A closing lemma on the interval. Inv. Math. 54 N2 (1979), 179187.

[26]L. Jonker and D. A. Rand . Bifurcations in one dimension, I: The non-wandering set. Inventions Math. 62 (1981), 347365.

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