Pisot numbers and strong negations

We introduce and study representation systems for the numbers in the unit interval [0, 1]. We call them ϕm-systems (where ϕm is a pseudo-golden ratio). With the aid of these representation systems, we define a family hm of strong negations and an increasing function gm which is the inverse of the generator of hm. The functions hm and gm are singular, and we study several properties; among which we calculate the Hausdorff dimensions of certain sets that are related to them. Finally, we prove that gm is an infinite convolution, and the sequence of coefficients in the Fourier series of its associated Stieltjes measure does not converge to zero.