Weakly attracting repellors for piecewise convex maps

  • Tomoki Inoue


We introduce the notion of a “weakly attracting repellor” in one dimensional dynamical systems, which is an unstable fixed point with the following property: the time average of ‘distance’ between the fixed point and the iterates of a pointx converges to 0 for almost every pointx. We study two classes of piecewise convex maps. One is a class of piecewise convex increasing maps including some intermittent dynamical systems, and the other is a class of cusp type maps which is related to the Lorenz equation with a critical parameter. We give some conditions for a piecewise convex map having a weakly attracting repellor. We also study the Lyapunov numbers of piecewise convex maps with weakly attracting repellors. The key to our proof is a δ-finite invariant measure.

Key words

weakly attracting repellor δ-finite invariant measure ergodicity intermittency Lorenz equation 


  1. [B-M] M. Benedicks and M. Misiurewicz, Absolutely continuous invariant measures for maps with flat tops. Publ. Math. IHES,69 (1989), 203–213.MATHMathSciNetGoogle Scholar
  2. [11] T. Inoue, Asymptotic stability of densities for piecewise convex maps. To appear in Ann. Polon. Math.Google Scholar
  3. [12] T. Inoue, Ergodic theorems for maps with indifferent fixed points. In preparation.Google Scholar
  4. [I-I] T. Inoue and H. Ishitani, Asymptotic periodicity of densities and ergodic properties for nonsingular systems. Hiroshima Math. J.,21 (1991), 597–620.MATHMathSciNetGoogle Scholar
  5. [L-M] A. Lasota and M.C. Mackey, Probabilistic Properties of Deterministic Systems. Cambridge Univ. Press, 1984.Google Scholar
  6. [L-Y] A. Lasota and J.A. Yorke, Exact dynamical systems and the Frobenius-Perron operator. Trans. Amer. Math. Soc.,273 (1982), 375–384.MATHCrossRefMathSciNetGoogle Scholar
  7. [Ma] P. Manneville, Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems. J. Physique,41 (1980), 1235–1243.CrossRefMathSciNetGoogle Scholar
  8. [Mi] J. Milnor, On the concept of attractor. Comm. Math. Phys.,99 (1985), 177–195.MATHCrossRefMathSciNetGoogle Scholar
  9. [P] G. Pianigiani, First return map and invariant measures. Israel J. Math.,35 (1980), 32–48.MATHCrossRefMathSciNetGoogle Scholar
  10. [P-Y] G. Pianigiani and J.A. Yorke, Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc.,252 (1979), 351–366.MATHCrossRefMathSciNetGoogle Scholar
  11. [T1] M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points. Israel J. Math.,37 (1980), 303–314.MATHCrossRefMathSciNetGoogle Scholar
  12. [T2] M. Thaler, Transformations on [0,1] with infinite invariant measures. Israel J. Math.,46 (1983), 67–96.MATHCrossRefMathSciNetGoogle Scholar
  13. [W] P. Walters, An Introduction to Ergodic Theory. GTM 79, New York-Heidelberg-Berlin, Springer, 1982.Google Scholar
  14. [Y-Y] J.A. Yorke and E.D. Yorke, Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model. J. Statist. Phys.,21 (1979), 263–277.CrossRefMathSciNetGoogle Scholar

Copyright information

© JJIAM Publishing Committee 1992

Authors and Affiliations

  • Tomoki Inoue
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceHiroshima UniversityHiroshimaJapan

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