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

Let f be a continuous map of the circle S¹ or the interval I into itself, piecewise C¹ 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)^nk periodic points of period n_k with large derivative along the period, |(f^nk) ′'| > e ^(h-ε)nk 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.