Asymptotic likelihood of chaos for smooth families of circle maps

We consider a smooth two-parameter family $f_{a,L}\colon\theta\mapsto \theta+a+L\Phi(\theta)$ of circle maps with a finite number of critical points. For sufficiently large $L$ we construct a set $A_L^{(\infty)}$ of $a$-values of positive Lebesgue measure for which the corresponding $f_{a,L}$ exhibits an exponential growth of derivatives along the orbits of the critical points. Our construction considerably improves the previous one of Wang and Young for the same class of families, in that the following asymptotic estimate holds: the Lebesgue measure of $A_L^{(\infty)}$ tends to full measure in $a$-space as $L$ tends to infinity.