dynamics in increasingly general families of smooth one-dimensional maps [@BC1; @BC2; @R; @TTY; @T1; @T2; @WY1]. Families of maps with critical and singular sets were studied in [@LT; @LV; @PRV]. One key aspect of the singularities of our maps that has no analogue in [@LT; @LV; @PRV] is that, returns to a neighborhood of the singular set can happen very frequently. The previous arguments seem not sufficient to deal with points like this. To avoid problems arising from the logarithmic singularities, and to get the asymptotic estimate on the measure of $\Delta$, we introduce new arguments:

-   our definition of bound periods (see Sect.\[s2.3\]) incorporates the recurrence pattern of the critical orbits to both $C$ and $S$. Thus, the resultant bound period partition depends on the parameter $a$, and is not a fixed partition, as is the case in [@BC1; @BC2];

-   to get the asymptotic estimate $|\Delta|\to1$ as $L\to1$, we need to adandon starting an inductive construction with small intervals around Misiurewicz parameters. Instead we start with a large parameter set (denoted by $\Delta_{N}$), which is a union of a finite number of intervals. This necessitates additional works on establishing uniform hyperbolicity outside of a neighborhood of $C$, which 