Shrinking targets in parametrised families

Abstract: We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdorff dimension of the set of parameters for which the orbit of a fixed point has a certain shrinking target property. This generalises several similar results for $\beta$-transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.