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Transactions of the American Mathematical Society

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Hyperbolicity properties of $ C\sp 2$ multi-modal Collet-Eckmann maps without Schwarzian derivative assumptions

Authors: Tomasz Nowicki and Sebastian van Strien
Journal: Trans. Amer. Math. Soc. 321 (1990), 793-810
MSC: Primary 58F08; Secondary 58F13
MathSciNet review: 994169
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Abstract: In this paper we study the dynamical properties of general $ {C^2}$ maps $ f:[0,1] \to [0,1]$ with quadratic critical points (and not necessarily unimodal). We will show that if such maps satisfy the well-known Collet-Eckmann conditions then one has

(a) hyperbolicity on the set of periodic points;

(b) nonexistence of wandering intervals;

(c) sensitivity on initial conditions; and

(d) exponential decay of branches (intervals of monotonicity) of $ {f^n}$ as $ n \to \infty ;$

For these results we will not make any assumptions on the Schwarzian derivative $ f$. We will also give an estimate of the return-time of points that start near critical points.

References [Enhancements On Off] (What's this?)

  • [CE1] P. Collet and J.-P. Eckmann, Positive Liapounov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynamical Systems 3 (1983), 13-46. MR 743027 (85j:58092)
  • [CE2] -, Iterated maps on the interval as dynamical systems, Birkhäuser, Boston, Mass., 1980.
  • [Gu] J. Guckenheimer, Sensitive dependence on initial conditions for one dimensional maps, Comm. Math. Phys. 70 (1979), 133-160. MR 553966 (82c:58037)
  • [Ma] R. Mané, Hyperbolicity, sinks and measure in one dimensional dynamics, Comm. Math. Phys. 100 (1985), 495-524. MR 806250 (87f:58131)
  • [MS] W. de Melo and S. J. van Strien, A structure theorem in one-dimensional dynamics, Ann. of Math. (2) (1989). MR 997312 (90m:58106)
  • [Mi] M. Misiurewicz, Absolutely continuous measure for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 17-51. MR 623533 (83j:58072)
  • [No1] T. Nowicki, On some dynamical properties of $ S$-unimodal maps on an interval, Fund. Math. 126 (1985), 27-43. MR 817078 (87e:58173)
  • [No2] -, Symmetric $ S$-unimodal properties and positive Liapounov exponents, Ergodic Theory Dynamical Systems 5 (1985), 611-616. MR 829861 (87j:58059)
  • [No3] -, Positive Liapounov exponents of critical value of $ S$-unimodal mappings imply uniform hyperbolicity, Ergodic Theory Dynamical Systems 8 (1988), 425-435. MR 961741 (90c:58100)
  • [NS] T. Nowicki and S. van Strien, Absolutely continuous invariant measures for $ {C^2}$ unimodal maps satisfying the Collet-Eckmann conditions, Invent. Math. 93 (1988), 619-635. MR 952285 (89j:58068)
  • [Str1] S. van Strien, Smooth dynamics on the interval, New Directions in Dynamical Systems, (T. Bedford and J. Swift, Eds.), Cambridge Univ. Press, 1988, pp. 57-119. MR 953970 (89m:58125)
  • [Str2] -, Hyperbolicity and invariant measures for general $ {C^2}$ interval maps satisfying the Misiurewicz condition, Comm. Math. Phys. (to appear).

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