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

Fundamenta Mathematicae (1998)

• Volume: 157, Issue: 2-3, page 245-254
• ISSN: 0016-2736

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## Abstract

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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 ${e}^{\left(h-\epsilon \right){n}_{k}}$ periodic points of period ${n}_{k}$ with large derivative along the period, $|{\left({f}^{{n}_{k}}\right)}^{\text{'}}|>{e}^{\left(h-\epsilon \right){n}_{k}}$ for some subsequence ${n}_{k}$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $\left(1-\epsilon \right){e}^{hn}$ such points.

## How to cite

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Katok, A., and Mezhirov, A.. "Entropy and growth of expanding periodic orbits for one-dimensional maps." Fundamenta Mathematicae 157.2-3 (1998): 245-254. <http://eudml.org/doc/212289>.

@article{Katok1998,
abstract = {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, $|(f^\{n_k\})^\{\prime \}| > e^\{(h-ε)n_k\}$ for some subsequence $\{n_k\}$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1-ε) e^\{hn\}$ such points.},
author = {Katok, A., Mezhirov, A.},
journal = {Fundamenta Mathematicae},
keywords = {circle ; piecewise monotone; positive entropy; critical points},
language = {eng},
number = {2-3},
pages = {245-254},
title = {Entropy and growth of expanding periodic orbits for one-dimensional maps},
url = {http://eudml.org/doc/212289},
volume = {157},
year = {1998},
}

TY - JOUR
AU - Katok, A.
AU - Mezhirov, A.
TI - Entropy and growth of expanding periodic orbits for one-dimensional maps
JO - Fundamenta Mathematicae
PY - 1998
VL - 157
IS - 2-3
SP - 245
EP - 254
AB - 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, $|(f^{n_k})^{\prime }| > e^{(h-ε)n_k}$ for some subsequence ${n_k}$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1-ε) e^{hn}$ such points.
LA - eng
KW - circle ; piecewise monotone; positive entropy; critical points
UR - http://eudml.org/doc/212289
ER -

## References

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