Circularly invariant uniformizable probability measures for linear transformations

In this paper, we prove a threshold result on the existence of a circularly invariant uniformizable probability measure (CIUPM) for linear transformations with non-zero slope on the line. We show that there is a threshold constant c depending only on the slope of the linear transformation such that there exists a CIUPM if and only if its support has a diameter at least as large as c. Moreover, the CIUPM is unique up to translation if the diameter of the support equals c.