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.
@article{bwmeta1.element.bwnjournal-article-cmv84i1p229bwm, author = {Gerhard Keller}, title = {A note on dynamical zeta functions for S-unimodal maps}, journal = {Colloquium Mathematicae}, volume = {84/85}, year = {2000}, pages = {229-233}, zbl = {0957.37020}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p229bwm} }
Keller, Gerhard. A note on dynamical zeta functions for S-unimodal maps. Colloquium Mathematicae, Tome 84/85 (2000) pp. 229-233. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv84i1p229bwm/
[000] [1] V. Baladi and G. Keller, Zeta-functions and transfer operators for piecewise monotone transformations, Comm. Math. Phys. 127 (1990), 459-478. | Zbl 0703.58048
[001] [2] V. Baladi, Periodic orbits and dynamical spectra, Ergodic Theory Dynam. Systems 18 (1998), 255-292. | Zbl 0915.58088
[002] [3] H. Bruin and G. Keller, Equilibrium states for S-unimodal maps, ibid. 18 (1998), 765-789. | Zbl 0916.58020
[003] [4] G. Keller and T. Nowicki, Fibonacci maps re(al)visited, ibid. 15 (1995), 99-120. | Zbl 0853.58072
[004] [5] W. de Melo and S. van Strien, One-Dimensional Dynamics, Springer, 1993. | Zbl 0791.58003
[005] [6] T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math. 132 (1998), 633-680. | Zbl 0908.58016
[006] [7] Y. Oono and Y. Takahashi, Chaos, external noise and Fredholm theory, Progr. Theor. Phys. 63 (1980), 1804-1807. | Zbl 1060.37501
[007] [8] R. Remmert, Theory of Complex Functions, Grad. Texts in Math. 122, Springer, New York, 1991.
[008] [9] D. Ruelle, Analytic completion for dynamical zeta functions, Helv. Phys. Acta 66 (1993), 181-191. | Zbl 0829.58033
[009] [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. | Zbl 0541.58030