Hyperbolicity properties of 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 maps 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 as

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

**[CE1]**P. Collet and J.-P. Eckmann,*Positive Liapunov exponents and absolute continuity for maps of the interval*, Ergodic Theory Dynam. Systems**3**(1983), no. 1, 13–46. MR**743027**, 10.1017/S0143385700001802**[CE2]**-,*Iterated maps on the interval as dynamical systems*, Birkhäuser, Boston, Mass., 1980.**[Gu]**John Guckenheimer,*Sensitive dependence to initial conditions for one-dimensional maps*, Comm. Math. Phys.**70**(1979), no. 2, 133–160. MR**553966****[Ma]**Ricardo Mañé,*Hyperbolicity, sinks and measure in one-dimensional dynamics*, Comm. Math. Phys.**100**(1985), no. 4, 495–524. MR**806250****[MS]**W. de Melo and S. van Strien,*A structure theorem in one-dimensional dynamics*, Ann. of Math. (2)**129**(1989), no. 3, 519–546. MR**997312**, 10.2307/1971516**[Mi]**Michał Misiurewicz,*Absolutely continuous measures for certain maps of an interval*, Inst. Hautes Études Sci. Publ. Math.**53**(1981), 17–51. MR**623533****[No1]**Tomasz Nowicki,*On some dynamical properties of 𝑆-unimodal maps on an interval*, Fund. Math.**126**(1985), no. 1, 27–43. MR**817078****[No2]**Tomasz Nowicki,*Symmetric 𝑆-unimodal mappings and positive Liapunov exponents*, Ergodic Theory Dynam. Systems**5**(1985), no. 4, 611–616. MR**829861**, 10.1017/S0143385700003199**[No3]**Tomasz Nowicki,*A positive Liapunov exponent for the critical value of an 𝑆-unimodal mapping implies uniform hyperbolicity*, Ergodic Theory Dynam. Systems**8**(1988), no. 3, 425–435. MR**961741**, 10.1017/S0143385700004569**[NS]**T. Nowicki and S. van Strien,*Absolutely continuous invariant measures for 𝐶² unimodal maps satisfying the Collet-Eckmann conditions*, Invent. Math.**93**(1988), no. 3, 619–635. MR**952285**, 10.1007/BF01410202**[Str1]**Sebastian van Strien,*Smooth dynamics on the interval (with an emphasis on quadratic-like maps)*, New directions in dynamical systems, London Math. Soc. Lecture Note Ser., vol. 127, Cambridge Univ. Press, Cambridge, 1988, pp. 57–119. MR**953970****[Str2]**-,*Hyperbolicity and invariant measures for general**interval maps satisfying the Misiurewicz condition*, Comm. Math. Phys. (to appear).

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DOI:
https://doi.org/10.1090/S0002-9947-1990-0994169-6

Article copyright:
© Copyright 1990
American Mathematical Society