phenomenon [@T] which is unknown to occur for smooth one-dimensional maps with critical points.

Our main result states that nonuniform expansion prevails for “most" parameters, provided $L\gg1$. Let $\lambda=10^{-3}$ and let $|\cdot|$ denote the one-dimensional Lebesgue measure.

For all large $L$ there exists a set $\Delta=\Delta(L) \subset [0,1)$ of $a$-values with $|\Delta|>0$ such that if $a \in
\Delta$ then for $f=f_a$ and each $c \in C$, $|(f^n)'(fc)|\geq L^{\lambda n}$ holds for every $n\geq 0$. In addition, $|\Delta|\to1$ holds as $L \to \infty$.

For the maps corresponding to the parameters in $\Delta$, our argument shows a nonuniform expansion, i,e, for Lebesgue a.e. $x\in S^1$, $\displaystyle{\varlimsup_{n\to\infty}}\frac{1}{n}\ln|(f_a^n)'x|\geq\frac{1}{3}\ln L.$ In addition, combining our argument with an argument in \[[@WY1] Sect.3\] one can construct invariant probability measures absolutely continuous with respect to Lebesgue measure (acips). The main difference from the smooth case is to bound distortions, which can be handled with Lemma \[dist\] in this paper. A careful construction exploiting the largeness of $L$ shows the uniqueness of acips and some advanced statistical limit theorems (See [@T]).

Since the pioneering work of Jakobson [@J], there have been quite a few number of papers over the last thirty years dedicated to proving the abundance of chaotic 