Some dynamical properties of S-unimodal maps

Tomasz Nowicki

Fundamenta Mathematicae (1993)

  • Volume: 142, Issue: 1, page 45-57
  • ISSN: 0016-2736

Abstract

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We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.

How to cite

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Nowicki, Tomasz. "Some dynamical properties of S-unimodal maps." Fundamenta Mathematicae 142.1 (1993): 45-57. <http://eudml.org/doc/211971>.

@article{Nowicki1993,
abstract = {We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.},
author = {Nowicki, Tomasz},
journal = {Fundamenta Mathematicae},
keywords = {-unimodal map; Collet-Eckmann map; invariant measure; Perron-Frobenius operator},
language = {eng},
number = {1},
pages = {45-57},
title = {Some dynamical properties of S-unimodal maps},
url = {http://eudml.org/doc/211971},
volume = {142},
year = {1993},
}

TY - JOUR
AU - Nowicki, Tomasz
TI - Some dynamical properties of S-unimodal maps
JO - Fundamenta Mathematicae
PY - 1993
VL - 142
IS - 1
SP - 45
EP - 57
AB - We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.
LA - eng
KW - -unimodal map; Collet-Eckmann map; invariant measure; Perron-Frobenius operator
UR - http://eudml.org/doc/211971
ER -

References

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  1. [BL] A. Blokh and M. Lyubich, Measurable dynamics of S-unimodal maps of the interval, Ann. Sci. École Norm. Sup. 24 (1991), 545-573. Zbl0790.58024
  2. [CE1] P. Collet and J.-P. Eckmann, Iterated Maps on the Interval as Dynamical Systems, Birkhäuser, Boston 1980. Zbl0458.58002
  3. [CE2] 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. Zbl0532.28014
  4. [K] G. Keller, Exponents, attractors and Hopf decomposition for interval maps, ibid. 10 (1990), 717-744. Zbl0715.58020
  5. [KN] G. Keller and T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for the Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), 31-69. Zbl0763.58024
  6. [L] F. Ledrappier, Some properties of absolutely continuous invariant measures on an interval, Ergodic Theory Dynamical Systems 1 (1981), 77-93. Zbl0487.28015
  7. [MS] W. de Melo and S. van Strien, One Dimensional Dynamic, book manuscript. 
  8. [M] M. Misiurewicz, Absolutely continuous invariant measures for certain maps of an interval, Publ. Math. I.H.E.S. 53 (1981), 17-51. Zbl0477.58020
  9. [N1] T. Nowicki, On some dynamical properties of S-unimodal maps of an interval, Fund. Math. 126 (1985), 27-43. Zbl0608.58030
  10. [N2] T. Nowicki, Symmetric S-unimodal mappings and positive Liapunov exponents, Ergodic Theory Dynamical Systems 5 (1985), 611-616. Zbl0615.28009
  11. [N3] T. Nowicki, A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity, ibid. 8 (1988), 425-435. Zbl0638.58021
  12. [NS1] T. Nowicki and S. van Strien, Hyperbolicity properties of multimodal Collet-Eckmann maps without Schwarzian derivative conditions, Trans. Amer. Math. Soc. 321 (1990), 793-810. Zbl0731.58021
  13. [NS2] T. Nowicki and S. van Strien, Absolutely continuous invariant measures for C 2 unimodal maps satisfying Collet-Eckmann conditions, Invent. Math. 93 (1988), 619-635. Zbl0659.58034
  14. [NS3] T. Nowicki and S. van Strien, Invariant measures exist under a summability condition for unimodal maps, ibid. 105 (1991), 123-136. Zbl0736.58030
  15. [Sz] B. Szewc, Perron-Frobenius operator in spaces of smooth functions on an interval, Ergodic Theory Dynamical Systems 4 (1984), 613-641. 
  16. [Y] L.-S. Young, Decay of correlations for certain quadratic maps, Comm. Math. Phys. 146 (1992), 123-138. Zbl0760.58030

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