Entropy and growth of expanding periodic orbits for one-dimensional maps

Katok, A.; Mezhirov, A.

Katok, A. ; Mezhirov, A.

Fundamenta Mathematicae, Tome 158 (1998), p. 245-254 / Harvested from The Polish Digital Mathematics Library

Résumé

Let f be a continuous map of the circle $S^{1}$ or the interval I into itself, piecewise $C^{1}$ , piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least $e^{(h - ε) n_{k}}$ periodic points of period $n_{k}$ with large derivative along the period, $| {(f^{n_{k}})}^{'} | > e^{(h - ε) n_{k}}$ for some subsequence $n_{k}$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1 - ε) e^{h n}$ such points.

Publié le : 1998-01-01

Zbl 0915.58025

EUDML-ID : urn:eudml:doc:212289

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     author = {A. Katok and A. Mezhirov},
     title = {Entropy and growth of expanding periodic orbits for one-dimensional maps},
     journal = {Fundamenta Mathematicae},
     volume = {158},
     year = {1998},
     pages = {245-254},
     zbl = {0915.58025},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv157i2p245bwm}
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Katok, A.; Mezhirov, A. Entropy and growth of expanding periodic orbits for one-dimensional maps. Fundamenta Mathematicae, Tome 158 (1998) pp. 245-254. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv157i2p245bwm/

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