A note on dynamical zeta functions for S-unimodal maps

Gerhard Keller

Colloquium Mathematicae (2000)

  • Volume: 84/85, Issue: 1, page 229-233
  • ISSN: 0010-1354

Abstract

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Let f be a nonrenormalizable S-unimodal map. We prove that f is a Collet-Eckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map.

How to cite

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Keller, Gerhard. "A note on dynamical zeta functions for S-unimodal maps." Colloquium Mathematicae 84/85.1 (2000): 229-233. <http://eudml.org/doc/210800>.

@article{Keller2000,
abstract = {Let f be a nonrenormalizable S-unimodal map. We prove that f is a Collet-Eckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map.},
author = {Keller, Gerhard},
journal = {Colloquium Mathematicae},
keywords = {dynamical zeta function; Collet-Eckmann map},
language = {eng},
number = {1},
pages = {229-233},
title = {A note on dynamical zeta functions for S-unimodal maps},
url = {http://eudml.org/doc/210800},
volume = {84/85},
year = {2000},
}

TY - JOUR
AU - Keller, Gerhard
TI - A note on dynamical zeta functions for S-unimodal maps
JO - Colloquium Mathematicae
PY - 2000
VL - 84/85
IS - 1
SP - 229
EP - 233
AB - Let f be a nonrenormalizable S-unimodal map. We prove that f is a Collet-Eckmann map if its dynamical zeta function looks like that of a uniformly hyperbolic map.
LA - eng
KW - dynamical zeta function; Collet-Eckmann map
UR - http://eudml.org/doc/210800
ER -

References

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  1. [1] V. Baladi and G. Keller, Zeta-functions and transfer operators for piecewise monotone transformations, Comm. Math. Phys. 127 (1990), 459-478. Zbl0703.58048
  2. [2] V. Baladi, Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems 18 (1998), 255-292. Zbl0915.58088
  3. [3] H. Bruin and G. Keller, Equilibrium states for S-unimodal maps, ibid. 18 (1998), 765-789. Zbl0916.58020
  4. [4] G. Keller and T. Nowicki, Fibonacci maps re(al)visited, ibid. 15 (1995), 99-120. Zbl0853.58072
  5. [5] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, 1993. Zbl0791.58003
  6. [6] T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math. 132 (1998), 633-680. Zbl0908.58016
  7. [7] Y. Oono and Y. Takahashi, Chaos, external noise and Fredholm theory, Progr. Theor. Phys. 63 (1980), 1804-1807. Zbl1060.37501
  8. [8] R. Remmert, Theory of Complex Functions, Grad. Texts in Math. 122, Springer, New York, 1991. 
  9. [9] D. Ruelle, Analytic completion for dynamical zeta functions, Helv. Phys. Acta 66 (1993), 181-191. Zbl0829.58033
  10. [10] Y. Takahashi, An ergodic-theoretical approach to the chaotic behaviour of dynamical systems, Publ. R.I.M.S. Kyoto Univ. 19 (1983), 1265-1282. Zbl0541.58030

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