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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, $|(f^{n_k})'| > e^{(h-ε)n_k}$ 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.