Ergodic properties of skew products withfibre maps of Lasota-Yorke type

We consider the skew product transformation T(x,y)= (f(x), ${\displaystyle T_{e\left(x\right)}}$) where f is an endomorphism of a Lebesgue space (X,A,p), e : X → S and ${\displaystyle {\left\{T_s\right\}}_{s\in S}}$ is a family of Lasota-Yorke type maps of the unit interval into itself. We obtain conditions under which the ergodic properties of f imply the same properties for T. Consequently, we get the asymptotical stability of random perturbations of a single Lasota-Yorke type map. We apply this to some probabilistic model of the motion of cogged bits in the rotary drilling of hard rock with high rotational speed.