number of iterates. One way to find these intervals is to first look for Misiuriewicz parameters, for which all critical orbits stay out of the bad sets under any positive iterate. We would then confine ourselves in small parameter intervals containing the Misiuriewicz parameters, and would eventually prove that the Misiuriewicz parameters are Lebesgue density points of the good parameter sets. This approach for initial set-ups, however, is with some drawbacks. First, for a one-parameter family of maps with multiple critical points, the Misiuriewicz parameters are relatively hard to find because of the need of controlling multiple critical orbits with one parameter. Though the argument in [@WY2] is readily extended to cover our family, we are nevertheless up to a hard start. Second, with the rest of the study confined in a small parameter interval containing a Misiuriewicz parameter, it is not clear how we could prove the global asymptotic measure estimate ($|\Delta| \to 1$ as $L \to \infty$) of the theorem.

An alternative route that is made possible by the approach of this paper is to start with a rather straight forward and relatively weak assumption. Let $\sigma = L^{-\frac16}$. Let $N$ be a large integer independent of $L$. 