On co-$\sigma$-porosity of the parameters with dense critical orbits for skew tent maps and matching on generalized $\beta$-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 $\beta$-transformations. As an application, we show that for the generalised $\beta$-transformation with the tribonacci number as slope, there is matching (i.e., $T^n(0)=T^n(1)$ for some $n \geq 1$) for Lebesgue-a.e. translation parameter.