On co-[formula omitted]-porosity of the parameters with dense critical orbits for skew tent maps and matching on generalised [formula omitted]-transformations

Abstract

We prove that the critical point and the point 1 have dense orbits for Lebesgue-a.e., parameter pairs in the two-parameter skew tent family and generalised β-transformations. As an application, we show that for the generalised β-transformation with the tribonacci number as slope, there is matching (i.e., Tn(0)=Tn(1) for some n≥1) for Lebesgue-a.e. translation parameter.